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Thrill U. THE PHYSICS AND MATH OF AMUSEMENT PARK RIDES Geometry © Copyrighted by Dr. Joseph S. Elias. This material is based upon work supported by the National Science Foundation under Grant No. 9986753. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. Dorney Park/Kutztown University Thrill U. - Geometry Introduction Welcome to Thrill U.! This set of mathematics activities focuses on Geometry. We believe there is something for everyone as the collection represents a breadth of adventures. Collectively, the geometry activities span a wide range of topics found in informal geometry, plane geometry, analytic geometry and foundations of trigonometry. Some activities require straightforward data collection and calculations, many require collaboration and some involve interdisciplinary considerations. Several of the activities can be easily adapted for special needs students, some can be adapted for use as followup activities, and still others are appropriate for schools with intensive scheduling programs. Each activity is preceded by an "Information Sheet" to help guide teachers in selecting appropriate activities for their particular group(s) of students. In addition to identifying objectives based on state and national mathematics standards, these pages provide a list of equipment needs and suggestions that may lead to the activity's successful completion. Teachers should feel welcome to adapt activities to the specific needs of their students. Teachers may request a “solution manual.” Contact Dr. Joseph S. Elias at [email protected]. Join us in May and challenge your students to experience geometry in action! Dr. Kathleen Dolgos Professor Emeritus College of Education Kutztown University Dr. Deborah Frantz Professor Emeritus College of Liberal Arts and Sciences Kutztown University Thrill U. Table of Contents Acknowledgments Page i Tips for Mathematics Teachers Page ii Things to Bring/Dorney Park Information Page iii Algebra Activities (separate manual) Pages 1 - 60 Geometry Activities Specific Rides: The Antique Carrousel Page 61 The Ferris Wheel Page 69 The Sea Dragon Page 75 Steel Force Page 81 White Water Landing – The Bridge Page 85 A Geometric Walk Page 91 Polygons in the Park Page 98 Rides That are Inscribed Page 103 The Fish Pond Page 110 Features in the Park: Dorney Park/Kutztown University Acknowledgments During the winter of 1997, area teachers of physics and mathematics, professionals from Dorney Park, and faculty from Kutztown University gave birth to Thrill U.: The Physics and Math of Thrill Rides. In May 1999, the Thrill U. - Physics Planning Committee presented physics activities in the form of Thrill U. The Thrill U. - Mathematics Planning Committee was established in the fall of 1999. The Committee began by creating algebra activities that would help teachers achieve goals set forth by the state and national “Mathematics Standards.” As a result, the mathematics component of Thrill U. consisted of algebra activities that were introduced in May 2001. Geometry activities have been developed and have been a part of the mathematics component since May 2002. Thrill U. is the culmination of effort and time of many people. Its existence would not have been possible without the collaborative efforts of: the professional staff at Dorney Park and Wildwater Kingdom; the administrators and academic faculty at Kutztown University; teachers who had taken students to the Park and provided feedback; and (most importantly) members of the planning committees. Each planning committee consists primarily of area high school teachers of physics or mathematics. The leadership and creativity of these teachers resulted in impressive sets of activities. Members of all planning committees have worn out shoes in the Park, endured days of inclement weather, fretted over success (or failure) of their students to complete preliminary versions of activities while in the "piloting" stages, spent many hours in meetings, and countless hours designing and editing the activities. In short, we admire and appreciate the efforts of all who have contributed to the success of the Thrill U. project. Geometry Planning Committee Mrs. Susan Barnett Ms. Karen Comegys Mrs. Terri Costenbader Dr. Kathleen Dolgos Dr. Joseph Elias Dr. Deborah Frantz Mrs. Rose Gadbois Mr. Keith Koepke Mr. Fran McGouldrick Mrs. Brenda Snyder Mr. Charles Waitkus Northwestern Lehigh High School William Allen High School Allentown Central Catholic High School Kutztown University Kutztown University Kutztown University Allentown Central Catholic High School Dorney Park and Wildwater Kingdom Allentown Central Catholic High School Kutztown University William Allen High School Tips for Mathematics Teachers i KUTZTOWN UNIVERSITY OF PENNSYLVANIA Think of this as an adventure! To help make a "stress-free" day at the Park, we have created this list of suggestions to guide you through your planning stage of Thrill U.. • Above and beyond all else, bring your sense of humor. Experienced teachers know that there will be mistakes. Allow students to have fun as well as complete your selection of activities. • While mathematics is an "exact" discipline, applications of mathematics are much "less exact." ALL measurements and collected data will have inherent errors. Accept it. • Please do not forget copies of activity sheets, equipment and supplies. You might also consider bringing a camcorder to record aspects of the rides for use in the classroom after Thrill U., or to use as introductory preparation for next year. • If your comfort level is low with orchestrating lab-type activities, consider consulting a science teacher for assistance with logistics. • Carefully peruse the complete list of activities and select those that will best fit the needs and abilities of your students. (That is, do not expect your students to complete all of these activities!) The difficulty levels are quite varied among the activities. Consider doing parts (but not all) of some activities. You may modify them, or assign groups of students to them. • Some activities take longer than others to complete. Keep in mind that it may be necessary to observe, ride, or take measurements several times in order to obtain good data. • As much as is feasible, introduce the students to the concepts to be studied during the weeks leading up to the event. Consider planning time in class for calculations and analysis during the days following the experience. • In our opinion, students who may be fearful of some rides should not be forced to ride. • Kutztown University students will serve as general assistants for you. They will be stationed at designated Thrill U. rides from approximately 10:00 A.M. to 12:00 P.M. Inform your students that they may ask the university students questions related to the activities. University students will help students discover the "answers," but will not give them answers. Instruct your students NOT to ask Dorney Park employees to give answers. • Teachers are welcome to utilize a designated grove at Dorney Park to chat with other teachers and members of the planning committees, or to use as a place for your students to work. Please do not leave equipment and other valuables unattended at the grove. Things to Bring We present this list for your convenience and hope that it helps make your day at the Park enjoyable as well as productive. You may wish to bring some or all of these items with you to Thrill U. ii KUTZTOWN UNIVERSITY OF PENNSYLVANIA • • • • • • • • • tickets for you, your students and your chaperones copies of your selection of activities, enough for your group pencils and paper stopwatches calculators (depends on activities: basic, basic with tangent key, or graphing) camcorder clipboards inclinometers or protractors with a plumb bob appropriate clothing and perhaps a change of clothing • • • • • • • sunscreen, hats, raincoats money for food, drinks, phone measuring tape or string masking tape backpacks or plastic bags to keep papers and equipment dry and together maps of the Park (can be picked up at the entrance to the Park) a good reserve of energy and enthusiasm for exploration Dorney Park Information General Information: (800) 551-5656 or Group Sales Information: (610) 395-2000 (610) 395-3724 For specific questions about ticket sales for Thrill U., call Matt Stoltzfus at (610) 3927607 or e-mail him at [email protected] Visit our website: Dorney Park Thrill U. iii KUTZTOWN UNIVERSITY OF PENNSYLVANIA The Antique Carrousel Information Sheet topics: central angle interior and exterior angles inscribed polygon sector of a circle arc length objectives: to use the number of sectors in various circles to determine the measure of central angles and number of sides of inscribed polygons. to examine the relationship between the number of sides of the inscribed polygons, the measure of each central angle, and the measure of each exterior angle of these polygons. to approximate the arc length of a sector of a circle and use this to calculate the radius of the circle using the formula arc length = (measure of central ÷360°) 2r equipment: pencil calculator (basic four function, non-graphing type) activity sheets extra paper 25-foot tape measure notes to the teacher: Prior to your visit to Dorney Park, you may want to introduce the Polygon Grid Activity. Introduce the vocabulary terms associated with polygons. Please consider that all polygons in this activity are regular polygons. In addition to the Antique Carrousel, this activity includes the circular rides Waveswinger, Enterprise and Apollo 2000. 61 KUTZTOWN UNIVERSITY OF PENNSYLVANIA The Antique Carrousel Please notice that there are two carousels at Dorney Park. The Antique Carrousel has 4 rows of horses mounted to the rotating platform. Be sure you are doing this activity at the Antique Carrousel. Before you get started on this activity, let’s define some terms. F Sector of a circle E A CAB is a central angle. B C D CBD is an exterior angle of the polygon inscribed in circle A. It is adjacent and supplementary to interior CBE. AF , AE , and arc FE bound sector FAE. A slice of pizza is a good illustration of a sector of a circle. 62 KUTZTOWN UNIVERSITY OF PENNSYLVANIA The Antique Carrousel Study questions for the Antique Carrousel: 1. Observe the structure that encloses the Antique Carrousel and decide how many sides are in this regular polygon. Calculate the measure of one central angle, the measure of one interior angle and the measure of one exterior angle for this polygon. Number of sides in polygon 2. Measure of one central Measure of one interior Measure of one exterior Now look at the Antique Carrousel. As the ride turns, count the number of sectors in the top of the carousel. Each sector is bounded by rows of lights. How many sectors are there? _________________ This is one sector of the ride. Complete the grid: Number of sides in polygon Measure of one central Measure of one interior Measure of one exterior 63 KUTZTOWN UNIVERSITY OF PENNSYLVANIA The Antique Carrousel 3. Now take a look at the fence around the Antique Carrousel. Support pillar Measure from here To here Measure the distance between two adjacent fence posts. ___________________ We will use this distance as an approximation of the length of the arc for the circular area bounded by the fence. We will assume that all fence sections are congruent and are the same width as each of the 8 pillars, which hold up the roof over the carousel. There are 40 fence sections and 8 pillars. Now that you know the arclength of 1/48th of this circle, we are going to find the length of the radius of the circle. a. First find the measure in degrees of the central angle for this 48-gon. Use π = 3.14 Measure of central =_____________ b. Find the radius by solving this for r. arclength = (measure of central 360) 2 r Radius = _____________ c. Area = r² Use to calculate the area of the circle. Area =_______________ d. Use circle. Circumference = 2 r to calculate the circumference of this Circumference =______________ 64 KUTZTOWN UNIVERSITY OF PENNSYLVANIA The Antique Carrousel 4. Now go to the Wave Swinger and complete the grid. Use this regular polygon Number of sides in polygon 5. Measure of one central Measure of one interior Measure of one exterior Complete a grid for the Enterprise. The radius of the Enterprise is 8.5 meters. Use this to find the arclength between each pair of consecutive radii. A car serves as an arc. Number of sides in polygon Measure of one central Measure of one exterior Arc length between radii 65 KUTZTOWN UNIVERSITY OF PENNSYLVANIA The Antique Carrousel 6. Complete a grid for the Apollo 2000 The connecting arms create the polygon you are studying. Make a sketch of it in this space. Why do the angle measures in this regular polygon have repeating decimals? Number of sides in polygon Measure of one central Measure of one exterior What makes this polygon different from the previous rides you’ve studied? Explain here: 66 KUTZTOWN UNIVERSITY OF PENNSYLVANIA The Antique Carrousel Polygon Grid Activity Please complete this activity prior to attending Dorney Park Thrill U. day. # sides # angles # ’s Sum of measures of interior ’s in polygon Measure of 1 interior in regular polygon Measure of 1 exterior in regular polygon Sum of measures of exterior ’s in polygon 3 5 6 7 8 9 10 12 14 16 20 Using the pattern above, fill in the rest of the table: 43 38 23 170 172.5 175 7.2 4 67 KUTZTOWN UNIVERSITY OF PENNSYLVANIA The Antique Carrousel If the number of sides is expressed as N, then write a formula for the number of triangles, the sum of the measures of the interior angles in a polygon, the measure of 1 interior angle in a regular polygon, the measure of 1 exterior angle in a regular polygon, and the sum of the measures of the exterior angles in a polygon. # sides # angles # ’s Sum of measures of interior ’s in polygon Measure of 1 interior in regular polygon Measure of 1 exterior in regular polygon Sum of measures of exterior ’s in polygon N 68 KUTZTOWN UNIVERSITY OF PENNSYLVANIA The Ferris Wheel Information Sheet topics: similar triangles circumference velocity perimeter and area of a polygon angle measures objectives: to estimate the diameter of a circle to determine the speed that a rider moves on the Ferris Wheel. to calculate the perimeter, area and other measurements pertinent to polygons. equipment: activity sheets pencil calculator stop watch ruler and string, or protractor with plumb and straw notes to the teacher: Students will need to estimate the diameter of the wheel of the Ferris Wheel. In order to determine this distance, students may use similar triangles or trigonometry. This is an important part of this activity. The activity cannot be completed without determining these diameter. If using the similar triangle method on the Ferris Wheel, a 6 inch ruler works better than a 12 inch ruler. You may wish to direct your students to give all linear measures in feet, and square measures in square feet. Special note: To successfully complete this activity, students are not required to ride the Ferris Wheel. 69 KUTZTOWN UNIVERSITY OF PENNSYLVANIA The Ferris Wheel PART 1: The thrill rating of this ride is __________. 1. Estimate the diameter of the wheel. This can be done using similar triangles or trigonometry. A. Similar Triangle Method: Stand at the curb across from the ride entrance. This point is approximately 47 feet from the wheel. Hold your ruler up as if to measure the diameter of the wheel. Stretch the string from the bottom of the ruler to your eye and measure this length. Use proportional sides of similar triangles to estimate the diameter of the wheel. (Note that the length of side B1C1 is greater than the diameter of the wheel.) C1 d C ruler line of sight A B string h = height (ground to eye level) h B1 10ft.8in. ground level B. Trigonometry Method: Stand at the curb across from the ride entrance. This point is approximately 47 feet from the wheel. Attach the plumb to the protractor at the vertex mark of the protractor. Attach the straw to the straight edge of the protractor. Use this device to determine the angle of elevation. Use the appropriate trigonometric function to estimate the diameter of the wheel. 2. Find the circumference of the wheel. 70 KUTZTOWN UNIVERSITY OF PENNSYLVANIA The Ferris Wheel 3. Determine the average time that it takes the wheel to make one complete revolution. Time at least three revolutions and find the average. 4. Using the results of #2 and #3, determine the velocity (speed) in feet per second that a rider moves when the wheel is in motion. 5. If the diameter were half as large, but the time to complete one revolution remains the same, will the rider move faster or slower on the new Ferris Wheel? How much faster or slower? 71 KUTZTOWN UNIVERSITY OF PENNSYLVANIA The Ferris Wheel PART 2: 1. Determine the average number of revolutions per minute (rpm) that the wheel makes. 2. Determine the angular velocity, , in radians per second. = rpm's x 2 60 3. Multiply the angular velocity by the radius of the wheel. 4. Compare this result with the answer to #4 of PART 1. What did you find? Why do you think that you got your result? 72 KUTZTOWN UNIVERSITY OF PENNSYLVANIA The Ferris Wheel PART 3: Look at the wheel. You should see an inscribed polygon. X C B D Z A O 1. Count the number of sides. _______________ 2. What special name applies to this polygon? 3. Is this polygon regular? ______________ 4. Find the measure of AOB. __________ 5. Find the measure of OAB. __________ 6. Find the measure of ABC. __________ 7. Radius OA = __________. 8. Apothem OZ = __________. 9. Find the perimeter of the polygon. 10. Find the area of the polygon. _______________ __________ __________ 73 KUTZTOWN UNIVERSITY OF PENNSYLVANIA The Ferris Wheel 11. DCX is called _______________. 12. Find the measure of DCX. 13. How does mAOB compare to mDCX? 14. What is the sum of the measures of the angles of the polygon? 15. What is the sum of the measures of the exterior angles (one at each vertex) of the polygon? __________ __________ ______________________________ __________ 74 KUTZTOWN UNIVERSITY OF PENNSYLVANIA The Sea Dragon Information Sheet topics: Pythagorean theorem equilateral and isosceles triangles congruent triangles corresponding parts of triangles right triangle trigonometry objective: to find the lengths of the stationary beams to find the degree measure of all of the angles formed by the support beams equipment: activity sheets pencil calculator (scientific or better) notes to the teacher: The best view of the structure of Sea Dragon is at the exit side of the ride. 75 KUTZTOWN UNIVERSITY OF PENNSYLVANIA The Sea Dragon The stationary support beams of the Sea Dragon form numerous triangles. Your job is to calculate the measures of the unknown beams as well as all of the angles within the stationary support structure. Below is a drawing of the support structure as viewed from the exit side of Sea Dragon. The solid lines represent actual beams and the dotted lines represent constructed lines needed for relationships and calculations. B E A 1. C D ABC is an equilateral triangle with side 46 ft. An equilateral triangle can also be classified as an a. AB = BC = mBAC = º mBCA = triangle. AC = º mABC = º NOTE: BD is an altitude of equilateral triangle ABC. Therefore BDA BDC because mBDA = mBDC = Consequently, ABD and CBD are . º. triangles. 76 KUTZTOWN UNIVERSITY OF PENNSYLVANIA The Sea Dragon b. Since ABC is equilateral and BD is an altitude, AD = and CD = . Explain: c. We can now calculate the length of BD using the Pythagorean Theorem on either ABD or CBD. (altitude)² + (base)² = (hypotenuse)² What information have we found that allows us to use the Pythagorean Theorem? Using either ABD or CBD, put the mathematical names of the sides into the Pythagorean Theorem. ( )² + ( )² = ( )² Replace the names of the sides with their measures from #1a and #1b and calculate the length of BD to two decimal places, showing all of your work. BD = 2. AED is a a. triangle. mEDA = º and ED = 20 ft. From #1b we know AD = Doing this calculation yields AE = . How can we calculate AE ? . 77 KUTZTOWN UNIVERSITY OF PENNSYLVANIA The Sea Dragon b. AED CED because Therefore CE = . because . c. To calculate the measure of EAD we can use a trigonometric ratio. Fill in the blanks for these ratios using side names from AED. sin(mEAD) = cos(mEAD) = tan(mEAD) = Replace the side names with their measures. sin(mEAD) = cos(mEAD) = tan(mEAD) = Find decimal values for the ratios above. sin(mEAD) = cos(mEAD) = tan(mEAD) = Use an inverse function on your calculator to find the angle measure. Fill in the blanks with the appropriate decimal values from above. Sin-1 = mEAD Cos-1 Tan-1 = mEAD mEAD = º (to the nearest degree) = mEAD 78 KUTZTOWN UNIVERSITY OF PENNSYLVANIA The Sea Dragon d. Show two different ways that you can now calculate mECD. mECD = e. Explain how you will find mAED and mCED. mAED = 3. º (to the nearest degree) º mCED = º AEB CEB because a. Use your answer from #1c and information given in #2a to calculate BE. (Show your work.) BE = b. Based on information found in #1a and #2c show how can we find mBAE. (Show your work.) mBAE = º The degree measure of BCE = because 79 KUTZTOWN UNIVERSITY OF PENNSYLVANIA The Sea Dragon c. How can we determine the measures of ABE and CBE? Find mABE and mCBE. mABE = d. º mCBE = º How can we determine the measures of BEA and BEC? Find mBEA and mBEC. mBEA = º m BEC = º 80 KUTZTOWN UNIVERSITY OF PENNSYLVANIA Steel Force Information Sheet topics: angle of depression right triangle trigonometry slope objective: to approximate and compare the measures of the angles of depression and the slopes of the first and second hills using right triangle trigonometry and the definition of slope equipment: activity sheets and pencils calculator ruler or tape measure notes to the teacher: Please remind students that their answers should include appropriate units of measure. 81 KUTZTOWN UNIVERSITY OF PENNSYLVANIA Steel Force PART 1: Approximate the angle of depression of the first hill. The vertical drop is given. The length of the hill can be estimated by counting the number of vertical supports on the track. A (top of first hill) angle of depression Count vertical supports to find this distance. vertical drop (200 feet) B 1. C (tunnel entrance) Find the number of supports from the top of first hill to the point where it enters the tunnel. (Hint: you can get a good view of these supports while standing in line at Thunderhawk.) The number of supports is _______________. 2. Calculate the length of the first hill. a. From the answer in #1 above, subtract 2. ____________ b. To find the approximate length of the first hill, multiply the number in #2a by 24 feet (distance between supports). This is the approximate length of the first hill. The length of the first hill is __________________________________. 3. Use trigonometry to find mBAC. mBAC is _______________. 82 KUTZTOWN UNIVERSITY OF PENNSYLVANIA Steel Force 4. Find the measure of the angle of depression. The measure of the angle of depression of the first hill is _______________. PART 2: Approximate and compare the angle of depression of the second hill to that of the first hill. D top of the second hill Count the number of supports to get this length. E F Pace off this distance. 1. point where the track crosses the water Find the horizontal distance between the top of the second hill and the point where the track crosses the water (EF). Stand on the midway at a point directly below the top of the second hill. Walk on that midway in the direction of Thunder Creek Speedway. Count the number of paces you take until you reach the point where the track crosses the water. The number of paces is _______________. 2. Measure the length of each pace. (Use the average of three paces to get an accurate measurement.) The length of one pace is _______________. 3. Calculate the horizontal distance (EF). The horizontal distance between the top of the hill and the point where the track crosses the water is _______________. 83 KUTZTOWN UNIVERSITY OF PENNSYLVANIA Steel Force 4. Count the number of supports from the highest point on the second hill to the point where the track crosses the water. The number of supports is _______________. 5. Calculate the length of the second hill. a. From the answer in #4 above, subtract 2. ____________ b. To find the approximate length of the second hill, multiply the number in #5a by 24 feet (distance between supports). This is the approximate length of the second hill. The length of the second hill is ______________. 6. Use trigonometry to calculate mEDF. mEDF is _______________. 7. Find the measure of the angle of depression of the second hill. The measure of the angle of depression of the second hill is _______________. 8. Describe the differences between the angles of depression for the two hills on Steel Force. 9. Calculate the slope of each hill. The slope of the first hill is ________________. The slope of the second hill is ______________. 10. Describe the differences in the slopes of the two hills. 84 KUTZTOWN UNIVERSITY OF PENNSYLVANIA White Water Landing – The Bridge Information Sheet topics: circle circumference as related to radius linear and degree measure of a circle simple right triangle trigonometry objective: to determine the relative location of the center of the circle to appreciate the relationship between the circumference and radius of a circle equipment: activity sheets sextant pencil calculator notes to the teacher: Students must construct a 20 ft. measuring string before arriving at the park. The string must have markings every 6 inches (½ ft.) and weighted on one end using fishing sinkers, washers, nuts, or the like. The 6-inch markings should be measured beginning from the outside of the weight, where the string is attached. Students must also construct a simple sextant in order to measure angles vertically. Materials for the sextant: protractor with one-degree markings, a straw, fishing line, and a weight. Tie the fishing line to the protractor at the center of the straight edge (or through the hole if available), tape the straw to the straight edge of the protractor, tie the weight to the other end of the line so it can swing freely from the center of the protractor. Please note: Due to safety precautions, students must ride this ride to gain access to the bridge. Remind students that all equipment must be secured when riding the ride. 85 KUTZTOWN UNIVERSITY OF PENNSYLVANIA White Water Landing – The Bridge Goal: Determine the location of the center of the circle, formed in part by the arched support of the center section of the bridge at White Water Landing. (See photo below). ARCHED SUPPORT 1. The arched support as labeled above is an arc of a circle. a. How many degrees are in a circle? ______ b. If the arched support has an arc measure of 12 degrees, what fractional part of the circle does the arched support represent? (Give your answer as a fraction and as a repeating decimal.) fraction______ decimal______________ 2. Find the distance (linear measure) from the top center of the arc to the water. To calculate this distance, you will be using your sextant. 86 KUTZTOWN UNIVERSITY OF PENNSYLVANIA White Water Landing – The Bridge Point X is a point along the fence just to the right of the lamppost (with the red dot). Point T is a point on the top of the arched support aligned with the middle of the top of the sign. Point B is a point at the bottom of the bridge below the middle of the bottom of the sign. Point W (marked in red) is a point on the boat track, at water level, directly below the bridge. T 104.87 B X W a. Hold the sextant level with the top of the fence, look through the eyepiece (straw), and locate point B. (A team member must make sure the fishing line is not swinging and is aligned with the 90 mark on the protractor part of the sextant, while you are looking at point B.) Raise the front of the sextant from the horizontal to locate point T at the same position in the eye piece where point B was. Have the team member read the degree measure on the sextant (to the nearest 1/2-degree) when you have T in the correct position. The measure of BXT = _______. (Make sure the line has stopped swinging before reading mBXT.) b. Following the same procedure as in #2a, locate point B (recall that the line must be still and aligned with 90), lower the front of the sextant from the horizontal to locate point W at the same position in the eye piece where point B was. The measure of BXW = _______. You will be using the trigonometric function tangent (TAN) on your calculator to find BT and BW. side opposite the angle Given: TAN (angle) = side adjacent to the angle and BX = 104.87 ft., the distance from the fence to the bridge. 87 KUTZTOWN UNIVERSITY OF PENNSYLVANIA White Water Landing – The Bridge c. Find BT using the tangent. TAN (mBXT) = BT BX TAN (mBXT) = BT 104.87 ft. (Use the measure of BXT from #2a to continue.) TAN (mBXT) = BT 104.87 ft. BT = (104.87 ft.) TAN (mBXT) BT = _________ft. d. Calculate BW using the same procedure as in #2c. BW 104.87 ft. (Use the measure of BXW from #2b to continue.) TAN (mBXW) = TAN (mBXW) = BW 104.87 ft. BW = (104.87 ft.) TAN (mBXW) BW = ________ft. Using the information from #2c and #2d, how can you determine the distance from the top center of the arc to the water?__________________________________________________________. TW (arc to the water) = ___________ft. 3. Find the circumference of the circle formed by the arc of the arched support. a. In #1, we determined the fractional part of the circle to be_____________ and the decimal part of the circle that the arc represented to be________________. 88 KUTZTOWN UNIVERSITY OF PENNSYLVANIA White Water Landing – The Bridge b. Determine the linear measure of the arc using your measuring string. Measure along the top of the arc from the red dot to either end of the arc. (Measure to the nearest 1/4 of a foot.) length of ½ of the arc = _________ ft. c. The entire length of the arc is approximately _________ft. d. Based on measurements from #2d and #3c and without doing any calculations, where would you hypothesize (educated guess) the center of the circle to be located? (above the water, underwater but above the ground, or underground) _______________________________ __ Explain the reasoning for your conclusion_______________________________ _________________________________________________________________ _________________________________________________________________. e. Using the information in #3a and #3c, the circumference of the circle is __________ft. Show work here: 4. Find the radius of the circle. a. In #2d, we found the distance from the top center of the arc to the water line to be _________ft. and the depth of the pool is 2.5 ft. b. The circumference of a circle in terms of the radius is calculated by the formula C = ___________. In #3e, we found the circumference to be ________ft. How can we calculate the radius based on the circumference? _______________ _________________________________________________ ___. c. Doing this calculation yields a radius of _________ft. 89 KUTZTOWN UNIVERSITY OF PENNSYLVANIA White Water Landing – The Bridge 5. Based on your calculation in #4c and measurements in #4a, where is the center of the circle located? (above the water, underwater but above the ground, or underground) _________________________________________________________________. Explain your answer: _________________________________________________________ _______________________________________________________________________. a. Does this answer match your hypothesis in #3d? _____________ b. If not, why do you think your guess was wrong? ____________________________________ ___________________________________________________________ __. 90 KUTZTOWN UNIVERSITY OF PENNSYLVANIA A Geometric Walk Information Sheet topics: recognition of geometric shapes area of square, trapezoid, triangle, circle, rectangle circumference of a circle volume ratio objectives: to use appropriate formulas for area, volume and circumference to identify geometric shapes in a real world situation equipment: activity sheets pencil 50-foot tape measure basic calculator notes to the teacher: This is a good activity for working in pairs. Students need to record and measure as a team. When working with the outer rim of bricks, use two rows of bricks as a border. In other words, the identical 4 geometric shapes do not overlap. KUTZTOWN UNIVERSITY OF PENNSYLVANIA Special thanks to Sara Bechtel, Ken Eicheldinger and Angela Xander for PART 2 91 A Geometric Walk As you walk through the main gate of the Park, you will enter the plaza situated between the gate and the Antique Carrousel. PART 1: Looking down as you walk into the plaza, you will see four identical geometric shapes each of which look like Figure 1 below. The shaded portions indicate the location of bricks. Do not include the geometric shapes that have a center planter. Figure 1 - The Plaza Shape A B H G E F D C 1. Name the different geometric shapes that can be found in this section of the plaza walkway. 2. Find the areas of the following geometric figures found on the plaza walkway. Do not include the outer rim of bricks in your calculations. a. Area of rectangle ABCD b. Area of trapezoid DEFC c. Area of square EFGH d. Area of triangle ABD KUTZTOWN UNIVERSITY OF PENNSYLVANIA Special thanks to Sara Bechtel, Ken Eicheldinger and Angela Xander for PART 2 92 A Geometric Walk 3. Knowing the areas of the geometric figures in question #2, answer the following: a. Use the fact that the area of a geometric figure is equal to the sum of the areas of its parts to show that the area of rectangle ABCD is equal to the sum of the areas of its parts. b. Remember that there are four geometric shapes on the entire plaza. What is the total area of all the trapezoids in all four figures? Check your answer by using the areas of the rectangle and square. 4. The rectangular shaped figures on the walkway are surrounded by and include rectangular and square bricks. a. Find the area of one of the rectangles that includes the outer border of bricks (Figure 1). b. Find the ratio of the number of rectangular bricks to the number of square bricks surrounding and included in one of the four rectangles in the plaza. KUTZTOWN UNIVERSITY OF PENNSYLVANIA Special thanks to Sara Bechtel, Ken Eicheldinger and Angela Xander for PART 2 93 A Geometric Walk c. In #2, the areas of the rectangle ABCD and the square EFGH were found. Use these areas to find the total area of the shaded regions in Figure 1. d. Use the ratio found in #4b and your answer in #4c to determine the area covered by the rectangular bricks in the shaded regions in Figure 1. e. Use the ratio found in #4b and your answer in #4c to determine the area covered by the square bricks in the shaded regions in Figure 1. f. Find the area covered by the square bricks by using the measurements for the size of the brick. Also find the area covered by the rectangular bricks using the measurements for the size of the brick. g. Compare the answers from parts d, e and f above. Reflect on your answers. KUTZTOWN UNIVERSITY OF PENNSYLVANIA Special thanks to Sara Bechtel, Ken Eicheldinger and Angela Xander for PART 2 94 A Geometric Walk PART 2 (probability): What is the percent probability of a bean bag being drop from a helicopter landing on one of the following patterns? Hint: use the following formula P(target area) = area of target / area of total a. The square EFGH b. The trapezoid ABGH c. Any trapezoid d. The brown brick boarder of the plaza shape PART 3: If you walk further down the walkway past the Antique Carrousel, you will see several rectangular planters containing trees and mulch on the main Midway. Look at the planters in line with the fountain at the center of the Midway. 1. Find the area of the first rectangular planter that is not a square. Use the outside edge of the concrete to measure the sides. Do not include any bricks surrounding the planter. KUTZTOWN UNIVERSITY OF PENNSYLVANIA Special thanks to Sara Bechtel, Ken Eicheldinger and Angela Xander for PART 2 95 A Geometric Walk 2. Find the area of the section of the rectangular bed that contains mulch. Explain the procedure to find the area covered by only mulch. (Assume that the area to be covered is clear of all electrical boxes, trees, etc.) 3. Given that mulch can be purchased by the cubic yard, estimate the number of cubic yards of mulch that would be needed to fill one rectangular bed. (One cubic yard of mulch covers approximately 80 to 100 square feet if spread 3 inches deep.) Extension for Experts: 4. If the mulch can only be ordered by the bag, how many bags would be needed? One bag of mulch contains 3 cubic feet of mulch. KUTZTOWN UNIVERSITY OF PENNSYLVANIA 96 A Geometric Walk PART 4: In the middle of the main Midway, you will find a circular fountain filled with water. The fountain’s structure contains a middle circular concrete island. The six outer segments are flower beds. Figure 2 - The Fountain 1. Find the area and circumference of the circle formed by the outer edge of the fountain given that the radius of the circle is 97 inches. Area: _______ sq. inches Circumference: _______ inches 2. Find the area (i.e. the surface area) of the part of the fountain that holds water. The middle circular island has a radius of 31 inches. The distance from the edge of the circular island to the interior wall of the fountain is 54 inches. This geometric shape looks like a doughnut. 3. The usual depth of the water in the fountain is 8.5 inches. Find the volume of water needed to fill the fountain to this depth. KUTZTOWN UNIVERSITY OF PENNSYLVANIA 97 Polygons in the Park Information Sheet topics: classification of a wide variety of polygons identification of convex, concave, regular and non-regular polygons objectives: to recognize and name polygons inherent in the Park to accurately classify triangles and quadrilaterals to recognize characteristics of polygons (convex, concave, regular, etc.) to order the measures of areas of triangles and quadrilaterals to sketch polygons equipment: activity sheets and pencils notes to the teacher: Please keep in mind that this activity is not an activity on measurement. Students will be unable to obtain measurements of angles and lengths of sides for most of the polygons. Their answers for columns 3, 4, and 5 on the “Answer Grid” will be based on observations and reasoning skills. For questions with multiple answers (i.e. #4 and #13), please allow for different arrangements of the answers. For responses about the polygon in PART 1, #11, students should respond as though the hole were not there. KUTZTOWN UNIVERSITY OF PENNSYLVANIA 98 Polygons in the Park PART 1: The goal of this activity is to be able to accurately describe many different polygons that can be found in Dorney Park. (Please read through PART 2 before you start PART 1.) Place all of your answers on the grid that is provided at the end of this activity. For each of the following, a. identify (by its mathematical name) the polygon that is being described, (If you see a polygonal region, identify the polygon that describes its boundary.) b. draw a sketch of what you see; c. identify it as regular or not regular; and d. identify it as convex or concave. Your answers will be terms such as: pentagon, hexagon, decagon, dodecagon, icosagon, 36-gon, triangle, quadrilateral, etc. If you find a triangle or a quadrilateral, list as many names for it as you can: isosceles, equilateral, right, acute, obtuse, rectangle, square, parallelogram, trapezoid, or rhombus. 1. The base (bottom) of a car on the Ferris Wheel has this polygonal shape. 2. The metal bars that join the supports between the swings on Charlie Brown's Swing form this polygonal shape. 3. At Musik Express, each letter of the sentence "MIT MUSIK GEHT ALLES BESSER" is written on one of these. 4. The center structure on the wheel of Musik Express is comprised of these three polygonal regions. 5. The base of the center support of the Scrambler has this shape. 6. The roofs on the cars on the Enterprise, laid end to end, form this polygon. 7. At the top of the Dominator, in a plane parallel to the ground, there is a polygon that "connects" the three vertical supports. Name the polygon. 8. The roof of Patio Pizza is decorated with black and white polygons of this type. 9. This polygon can be viewed on the ground (made from red/brown bricks) and is the smallest polygon that surrounds the Antique Carrousel. KUTZTOWN UNIVERSITY OF PENNSYLVANIA 99 Polygons in the Park 10. Above the windows at the SUBWAY sandwich shop, there are polygonal signs that say "PAY HERE" or "ORDER HERE." What is the shape of each sign? 11. On the center support of the Scrambler, this polygonal region has a circular hole in it. (When working with this polygon, ignore the hole.) 12. At the Funzone Arcade, there are two different shapes that make up the three-piece emblem above the neon "FUNZONE" sign. Identify both polygons. 13. The arms that hold the cars on Apollo 2000 are connected to each other at the top of the ride. In a plane parallel to the ground, name the polygon that is formed by the connections of all the arms of the cars. 14. The perimeter of the pavilion in Camp Snoopy has this shape. PART 2: 1. Arrange the areas of all of the regions enclosed by triangles that were found in PART 1 from smallest to largest. Do this by listing the numbers of the objects. (Do you really need any formulas or computations of areas?) 2. Arrange the areas of all of the regions enclosed by quadrilaterals that were found in PART 1 from smallest to largest. Do this by listing the numbers of the objects. (Do you really need any formulas or computations of areas?) KUTZTOWN UNIVERSITY OF PENNSYLVANIA 100 Polygons in the Park Answer Grid for Polygons in the Park Problem number and Attraction 1. Ferris Wheel Names for the polygons Sketch Regular or not regular? Concave or convex? 2. Charlie Brown’s Swing 3. Musik Express (letters) 4a. Musik Express (center) 4b. Musik Express (center) 4c. Musik Express (center) 5. Scrambler (base) 6. Enterprise 7. Dominator 8. Patio Pizza KUTZTOWN UNIVERSITY OF PENNSYLVANIA 101 Polygons in the Park Problem number and Attraction 9. Antique Carrousel Names for the polygons Sketch Regular or not regular? Concave or convex? 10. Subway 11. Scrambler (hole) 12a. Funzone Arcade 12b. Funzone Arcade 13. Apollo 2000 14. Pavilion at Camp Snoopy KUTZTOWN UNIVERSITY OF PENNSYLVANIA 102 Rides That are Inscribed Information Sheet topics: properties of regular polygons (interior & exterior angles) area of regular polygon central angles area of circle area of sectors objectives: to use properties of regular polygons to use formulas for determining measures of interior and exterior angles to calculate lengths and areas associated with polygons inscribed in circles equipment: activity sheets pencil basic calculator notes to teacher: This activity contains three parts. PART 1 – Chart form investigation of interior and exterior angles. PART 2 – Question and answer investigation of regular polygons. PART 3 – Area investigation of inscribed polygons. PART 1: Regular Polygons – Interior & Exterior Angles Chart Attendance at the Park is needed to complete the Name of Ride column of the chart. If time is a factor, the other columns may be completed/calculated at a later time. http://mathforum.org/dr.math/ is a good source for the polygon names for 11 and up, if you are not going to use 11-gon, 12-gon, etc. PART 3: #1, 2, & 3 The circle provided has divisions of 5. KUTZTOWN UNIVERSITY OF PENNSYLVANIA 103 Rides That are Inscribed PART 1: Regular Polygons – Interior & Exterior Angles Chart Walk around the park. Look carefully at each ride. You should start to notice that most of the rides listed are circular in nature. An actual or visualized polygon is inscribed within the circle. For example an actual polygon exists on the Ferris Wheel. The vertices of the polygon are located at the ends of the spokes. The Tilt-A-Whirl’s polygon is visualized by joining segments from each seat’s pivot point. Examples of how the rides may appear (bird’s eye view) as inscribed polygons: KEY: 1. Actual structure Visualized structure Why do you think most of the rides have polygons with an even number of sides? __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ Instructions for Regular Polygons – Interior & Exterior Angles chart. a. Determine where the 10 rides listed below fit into the chart. Ride/Structure list: Antique Carrousel Apollo 2000 Dominator Ferris Wheel Monster Musik Express Pavilion near Fossil Find and Snoopy Bounce Scrambler (configuration of seating area) Tilt-A-Whirl Wave Swinger. b. c. d. Fill in the Name of Polygon column. You may need your math book for this. Place the # of Vertices in that column. Complete the Measure of each Exterior column, using your information from # of Vertices and Sum of the measures of Exterior ’s columns. Use the information in your chart to complete the last two columns. e. KUTZTOWN UNIVERSITY OF PENNSYLVANIA 104 Rides That are Inscribed Regular Polygons – Interior & Exterior Angles 3 360 Sum of the measures of Interior + Exterior 180 4 360 180 5 360 180 6 360 180 7 360 180 8 360 180 * 9 360 180 * 10 360 180 * 11 360 180 12 360 180 13 360 180 14 360 180 15 360 180 16 360 180 17 360 180 18 360 180 19 360 180 20 360 180 Name of Ride * * * * * # Sides Name of Polygon # of Vertices Sum of the measures of Exterior ’s Measure of each Exterior Measure of each Interior Sum of the measures of Interior ’s * See if you can find a ride or structure to match the number of sides. KUTZTOWN UNIVERSITY OF PENNSYLVANIA 105 Rides That are Inscribed PART 2: Answer the following questions about polygons: 1. What makes a polygon regular? ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ 2. Use the figures below. Draw the diagonals from one vertex to help complete the following chart: Number of sides 3 4 5 6 7 8 9 n Number of diagonals from one vertex Number of triangles created by diagonals Sum of the m’s in a Sum of the measures of the interior’s in the polygon 3. Write a formula for the sum of the measures of the interior angles of a convex polygon with n sides. ________________________________________________________________________ 4. Write a formula for the measure of each interior angle of a regular polygon with n sides. ________________________________________________________________________ 5. What is the sum of the measures of the exterior angles of a convex polygon (one angle at each vertex)? ________________________________________________________________________ 6. Write a formula for the measure of each exterior angle of a regular polygon with n sides. ________________________________________________________________________ KUTZTOWN UNIVERSITY OF PENNSYLVANIA 106 Rides That are Inscribed Select True or False. Support your decision with a brief explanation. 7. The sum of the measures of the interior angles of a polygon is always a multiple of 180. True or False Explanation: _____________________________________________ _____________________________________________________________________ _____________________________________________________________________ 8. There is a regular polygon in which the sum of the measures of the interior angles is 310. True or False Explanation: _____________________________________________ _____________________________________________________________________ _____________________________________________________________________ 9. As the number of sides of a regular polygon increase, the measure of each interior angle will decrease. True or False Explanation: _____________________________________________ _____________________________________________________________________ _____________________________________________________________________ 10. As the number of sides of a regular polygon increase, the sum of its exterior angle measures will also increase. True or False Explanation: _____________________________________________ _____________________________________________________________________ _____________________________________________________________________ 11. Each exterior angle of a regular hexagon is acute. True or False Explanation: _____________________________________________ _____________________________________________________________________ _____________________________________________________________________ KUTZTOWN UNIVERSITY OF PENNSYLVANIA 107 Rides That are Inscribed PART 3: In the circles provided, accurately inscribe the regular polygon represented by each of the following rides: Dominator (completed for you), Scrambler (seat configuration), and Monster. Draw at least one central angle and one apothem. For each, calculate the length of the apothem, perimeter of the polygon, measure of one central angle, area of one triangle, and area of the polygon. 1. Dominator: Draw regular polygon ABC inscribed in circle O. Given: AB = 40 feet Find the following: A. apothem a of polygon ABC = __________ C B. perimeter of polygon ABC = __________ C. measure of central AOB = __________ D. area of AOB = _______ O E. area of polygon ABC = __________ A a B EXTENSION: F. area of sector AOB = __________ G. area of region bounded by AB and arc AB = __________ H. difference between the area of polygon ABC and circle O = __________ 2. Scrambler (seat configuration): Draw regular polygon ABCD inscribed in circle O. Given: diameter = 260 inches Find the following: A. apothem of polygon ABCD = __________ B. perimeter of polygon ABCD = __________ C. measure of central AOB = __________ D. area of AOB = __________ E. area of polygon ABCD = __________ EXTENSION: F. area of sector AOB = __________ G. area of region bounded by AB and arc AB = __________ H. difference between the area of polygon ABCD and circle O = __________ KUTZTOWN UNIVERSITY OF PENNSYLVANIA 108 Rides That are Inscribed 3. Monster: Draw regular polygon ABCDEF inscribed in circle O. Given: diameter = 42 feet Find the following: A. apothem of polygon ABCDEF = __________ B. perimeter of polygon ABCDEF = __________ C. measure of central AOB = __________ D. area of AOB = __________ E. area of polygon ABCDEF = __________ EXTENSION: F. area of sector AOB = __________ G. area of region bounded by AB and arc AB = __________ H. difference between the area of polygon ABCDEF and circle O = __________ KUTZTOWN UNIVERSITY OF PENNSYLVANIA 109 The Fish Pond Information Sheet Topics: Circumference Area Volume Trigonometry Objectives: To use the circumference of a circle to determine its radius. To calculate the area and volume of a cylinder given its dimensions. To calculate the height of an object using trigonometric ratios. Equipment: Pencil Calculator Activity sheets Tape measure String Inclinometer 110 KUTZTOWN UNIVERSITY OF PENNSYLANIA Special thanks to Sara Bechtel, Ken Eicheldinger and Angela Xander The Fish Pond The historic fish pond has been here since the 1920s. It is located near the entrance to Gasoline Alley. Part I: 1. Find the circumference of the pond. a. Walk around the fish pond to find the gate. It is near the hedge in the direction of Steel Force. b. Measure around the inside of the cement base of the pond from middle of post to middle of post on the gate. Use a string to do so, then measure length with a measuring tape. c. Measure one section on either side of the gate. Both are congruent, so we only need to measure once. Again, use the string and go from post to post around the inside of the cement base. d. The remaining sections of fence are congruent. Take a moment to walk around the pond and count the number of remaining fence sections. e. Measure one of these sections and record your results here. f. Calculate the circumference of the fish pond by adding all of your measurements. Remember to include all sections of fence and the gate. 111 KUTZTOWN UNIVERSITY OF PENNSYLANIA Special thanks to Sara Bechtel, Ken Eicheldinger and Angela Xander The Fish Pond 2. Calculate the area and volume of the pond. a. Recall that circumference of a circle is found using the formula C = 2 r. Using the circumference, find the radius of the pond. b. In order to calculate the area of the fish pond, we must take into consideration the fact that there is an island in the center. Given that the radius of the island is 10 feet, calculate the area of the pond. Recall that A = r2. c. Given that the depth of the pond measures 4 feet, calculate the volume of the pond. This will tell us the maximum amount of water that it could be contained within the pond. Don’t forget about the island in the center! 3. Find the height of the center tower. a. Using the top railing of the fence as your starting point, steady the inclinometer. Tilt it upward until the top of the tower is in your sight. Have a teammate read the angle of inclination. 112 KUTZTOWN UNIVERSITY OF PENNSYLANIA Special thanks to Sara Bechtel, Ken Eicheldinger and Angela Xander The Fish Pond b. Using the angle of inclination and the radius of the pond, calculate the height of the tower from the railing upward. Use the diagram as a reference. Height Angle Radius c. Because we started at the railing, instead of the ground, this is an incomplete height. You must now find the height from the ground to the top of the inclinometer as it rests on the top railing. d. Adding these two lengths together will give you the height of the tower. 113 KUTZTOWN UNIVERSITY OF PENNSYLANIA Special thanks to Sara Bechtel, Ken Eicheldinger and Angela Xander
