Download Lesson Plan: Cylinder Savvy

Lesson Plan: Cylinder Savvy
(With Teaching Comments)
Introduction Discussion: Teacher or student brings & shares example CD/DVD
cases/holders (Only share 1 or 2 designs so students can still be creative coming up
with a design of their own. You may also want to show them a tower of 30, 50, or
100 blank discs at some point and have them practice calculating the surface area
and volume of it. This will allow them to make comparisons with their own design
since a tower is a very efficient way to store a large number of discs. )
Questions:
• What are some features of the cases/holders? (How are the discs held in
place? Can you see disc titles? Is that important in its design?)
• What are some ways it could be improved? (quality, capacity, shape,
color, material, etc.)
• What is a prototype? (It’s a 3-D image or model of a product idea.)
• Why are prototypes necessary/useful? (Saves money to plan ahead and/or
build one before producing thousands you will want to put into the market.
Allows you to find defects, planning errors, etc. before investing in final
product costs.)
• When is precision important in measuring? In calculating? (When you
are drawing up something that actually needs to be built, and therefore, there
is little or no room for error in a product design. Estimating is okay, however,
for planning on how much material to purchase.)
Project Description:
Students will pretend they are an engineer/product designer creating a prototype
CD/DVD case. They will sketch a design for holding 12 discs securely. It must be in
the shape of a rectangular prism, triangular prism, cylinder, or other three
dimensional figure you can calculate the surface area and volume for. They must
make 3 versions of their design to meet the needs of customers. First, we will
practice finding the surface area and volumes of cylinders.
Teach & Practice Basic Skills:
Surface Area of a Cylinder Worksheet (Piece-by-piece description of the formula
for finding surface area; includes an example problem and practice problems)
Volume of a Cylinder Worksheet (description for finding volume; includes an
example problem and practice problems)
Demonstration/Exploration: Paper and Rice (Will get students thinking about how
surface area and volume are not related directly… one can change while the
other remains the same.)
Cylinder Savvy
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Lesson Project: Cylinder Savvy
Student Instructions Handouts for Designs 1, 2, and 3
Design 1 – Original Plan
Asks students to sketch a design for holding 12 discs securely. It must be in
the shape of a rectangular prism, triangular prism, cylinder, or other three
dimensional figure they can calculate the surface area and volume for.
Design 2 - Make it Taller
• adjust height to allow for more disc storage
• predict new surface area and volume
• find new surface area and volume
• note changes, discuss findings
Design 3 - Make it Narrower
• adjust radius/diameter of Design 1 to make it work for mini discs
• predict new surface area and volume
• find surface area and volume
• note changes, discuss findings
Extra Practice Activities:
Silo: Cylinder Change in Radius (Practice predicting & calculating how a
change in radius will affect the surface area and volume)
Candles: Cylinder Change in Height & Radius (Practice predicting &
calculating how change in height will affect surface area and volume;
addresses efficiency of product as well in a cost vs content analysis)
Assessment: 10 question WASL-like test
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TC-1
Pre-check Answer Key
1.
radius
diameter
circumference
2. Answers will vary; adjust for your teachings…
Perimeter of a rectangle
Add all the side lengths
Area of a rectangle
A=lxw
Surface Area of a rectangular prism Sum of the areas of the faces
Volume of a rectangular prism
V=lxwxh
Circumference of a circle
C=2xπxr
Area of a circle
A=πxrxr
3. C = 8π in. ≈ 25.12 in.
A = 16π sq. in. ≈ 50.24 sq. in.
4. C = 9 π ft. ≈ 28.26 ft.
A = 20.25 sq. ft. ≈ 63.59 sq. ft.
5. 1st circle: C = 2π in. ≈ 6.28 in.; A = π sq. in. ≈ 3.14 sq. in.
2nd circle: C = 4π in. ≈ 12.56in.; A = 4π sq. in. ≈ 12.56 sq. in.
Circumference changes by doubling (x2)
Area quadruples (x4) because radius is squared
If radius is 3, circumference will be triple (x3) 1st circle; area will be 3 squared (or 9
times) as large.
6. Original (4x5x8): SA = 184 sq. mm.; V = 160 cu. mm.
One dimension doubled (8x5x8): SA = 288 sq. mm.; V = 320 cu. mm.
Surface Area increased by 104 sq. mm. due to an added strip 4 mm. wide wrapping
around 4 sides (two of which were 5mm, two of which were 8mm). So we added
(4x5) +(4x8)+(4x5)+(4x8)=104.
Area doubled.
SA will add a strip 5mm wide that will be 8mm long on 4 sides, so 4x(5x8)=160 larger
than 288 sq.mm., or 448sq. mm.
Area will double again and will be 640cu.mm.
7. Prediction: Surface area will add four wrap-around strips to the front (2x7), top (2x2),
back (2x7), and bottom (2x2). This will add 14+4+14+4, or 36 sq. m. to the total
surface area. Volume has been doubled because we doubled the width of the
rectangular prism, and therefore doubled the amount of space the figure contains.
1st Figure SA = 64 sq.m.
V = 28 cu. m.
2nd Figure SA = 100 sq.m
V = 56 cu. m.
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TC-2 & TC-3 (see below)
Pre-lesson
Surface Area of a Cylinder
If students need to develop an understanding of the formula for finding surface area, the
Surface Area of a Cylinder worksheets (Forms A and B) are designed to help do that.
Questions posed ask students to look closely at the parts of a cylinder and how we can
determine the area of each of its surfaces. It is also a good review of vocabulary
associated with cylinders: lateral surface and base.
Use of Form A requires only the student worksheet, but Form B is for use with
manipulatives- some kind of cylinder object that can be cut apart and measured in
class. Your students will need to help you collect objects in advance. Select and use
only one of these forms, as they repeat information.
TC-2
Answer Key to Form A:
All answers use 3.14 for pi.
1. and 2. Check student work for understanding.
3. Area of each Circle = 50.24 sq. cm.
4. Circumference (Length of Lateral Surface) = 25.12 cm
5. Area of Lateral Surface = 150.72 sq. cm.
6.
Surface Name
Circle #1
Circle #2
Lateral Surface
Total Surface Area:
Formula for Area
πr 2
πr 2
2πr ⋅ h
2 ⋅ πr 2 + 2πr ⋅ h
Area
16π or 50.24 sq.cm.
16π or 50.24 sq.cm.
48π or 150.72 sq.cm.
80π or 251.2 sq. cm.
7. Work and answer shown below is on student worksheet. Students could leave
answer in form of pi and avoid the use of a calculator if desired.
2
SA = 2 ⋅ π (2) + 2π (2)(10)
= 2 ⋅ π (4 ) + 2π (20)
= 8π + 40π
= 48π ≈ 150.72in.2
8. 360 sq. ft. ≈ 1130.4 sq. ft. or
9. 24π sq. m ≈ 75.36 sq. m.
10. 66.5π sq,cm. ≈ 208.81 sq. cm
11. 85.50 sq. yd. ≈ 268.47 sq. yd
12. 250π sq.in. ≈ 785 sq. in.
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TC-3
Answer Key to Form B:
All answers use 3.14 for pi.
1. Base is always a CIRCLE on a cylinder. The “base” is not the face or the surface on
the bottom as one might assume. It will always be the significant face or surface which
gives the 3 dimensional solid its name. So, even when a cylinder is lying on its lateral
surface, its base is still a circle. The same is true for triangular prisms, for example. If a
triangular prism is drawn resting on a rectangular face, its base is still always one of the
triangles.
Numbers 2 through 5 – Answers will depend on manipulatives’ dimensions.
Check student work for understanding. If student cuts lateral surface as a shape other
than a rectangle, demonstrate for them how you can cut and paste the ends off to make
the shape a rectangle. If they find it to be a parallelogram, you can point out how the
length (as circumference) will still work, but your height must be that measured from the
original solid figure(this is because the formula for the area of a parallelogram is A=bxh,
where the height must be perpendicular to the base.)
The answer for number 6 shows how students could solve without using a calculator,
leaving the final answer in pi form. See below:
2
SA = 2 ⋅ π (2) + 2π (2)(10)
= 2 ⋅ π (4) + 2π (20)
= 8π + 40π
= 48π ≈ 150.72in.2
The work shown above is not following general order of operations, but does generate a
worthwhile discussion to have with the students. Because pi is written as a symbol, we
can treat it like a variable in a sense, doing all computations around it. Once all other
operations have been performed, it is mathematically sound to then substitute in 3.14
for pi and simplify for a decimal answer. However, because pi answers can be easier to
attain, you may want to have students put final answers in pi form.
In the following practice examples, most students will need repeated practice working
with such a long formula. Remind them of order of operations, and that after
substitution they should be squaring the radius before any multiplication or addition
takes place.
7. 1130.4 sq. ft. or 360 π sq. ft.
8. 75.36 sq. m. or 24 π sq. m
9. 208.81 sq. cm or 66.5 π sq. cm
10. 268.47 sq. yd. or 85.5 π sq. yd.
11. 785 sq. in. or 250 π sq. in.
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TC-4
Answer Key: Volume of a Cylinder
Pre-lesson
For students needing a review of volume, this supplemental worksheet is designed to
help with just that. Review with students why the volumes of all prisms and cylinders
can be found using the same formula. Once students can identify the “significant”
shape that serves as the base of the solid, they need only find its area and multiply by
the given height. Emphasize that identifying the base is key… even if a cylinder is
drawn on its side, the base is a circle (and triangles are always the base in a triangular
prism). So we must have a good understanding of which dimensions are which, despite
the positioning of our solid.
All answers use 3.14 for pi.
1. Answer is given on student form. See below:
V=Area of Base · Height
2
·
h
= π ⋅r
2
· (10)
= π (2)
= π4
· (10)
= 40π ≈ 125.6in.3
As students work, check that they are selecting radius and not diameter, and that they
understand how to correctly find the area of a circular base.
While using the formula is useful, students really need to understand where it comes
from and why it works. Check frequently that students know why the area of the base is
needed for finding volume. A question you may ask: How does height affect the volume
if the area of the base stays the same? Why?
Once they understand the formula’s origins, they are ready to use it more efficiently.
Keep an eye out for students who begin by multiplying, when they should always first
square the radius. Emphasize this in your review of their answers so they remember to
follow order of operations.
2. 880π cu. ft. ≈ 2512 cu. ft.
3. 16π cu.m ≈ 50.24 cu. m.
4. 73.5π cu. cm. ≈ 230.79 cu. cm.
5. 101.25π cu. yd. ≈ 317.925 cu. yd.
6. 500π cu. in. ≈ 1570 cu. in.
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TC-5
Teacher’s Version: Exploration/Demonstration: Paper & Rice
Pre-lesson
Goal: Demonstrate how changes in radius can affect the volume of cylinders without
affecting surface area.
Teacher can do in front of class, or students can actively discover findings in groups.
The following are directions for students in groups.
Teacher should try and encourage students to make their own predictions, not cluing
them in to the fact that the two designs will not have the same volume, despite their
equal surface areas.
Materials (for groups of 5 in a class of 35):
ƒ 8.5 x 11 inch sheets of paper
ƒ box lids or shallow boxes
• 5 bags of cheap rice (divided into 7 large re-sealable bags)
ƒ measuring cups (one cup works well)
• 7 scissors
• rulers
• tape
Directions for the student:
1. Have one person in your group cut the piece of paper exactly in half (hamburgerstyle). Label the half sheets A and B.
2. Measure and find the surface area of rectangle A.
3. Measure and find the surface area of rectangle B.
4. Tape A into a hotdog-style cylinder (longways).
5. Tape B into a hamburger-style cylinder (wider).
6. Predict below which cylinder will have greater volume.
7. While one student holds the cylinder in place measure and fill A with rice. What
is its approximate volume? ___________
8. Slip B over and around A. Lift A up and out of B so the rice begins to fill the
wider
cylinder.
What
occurred
once
A
was
completely
removed?
________________________
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Answer the following:
1. Surface Area of A: __________
2. Surface Area of B: __________
3. Prediction- Which cylinder will have greater volume? ________________
Why? _______________________________________
4. Why do you think the result occurs as it does?
________________________________________________________________
5. Can you sketch 2 cylinders for which the volume would be the same, but the
surface area would be different?
________________________________________________________________
Cylinder Savvy
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Schematics Needed for Cylinder Savvy Lesson
CD/DVD
Diameter = 12 cm, Thickness (height) = 0.5mm
Mini-CD
Diameter = 80 mm, Thickness (height) = 0.5mm
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Disc Storage Websites:
1. www.sleevetown.com/dvd-case.shtml
2. http://www.caselogic.com/search/index.cfm?Ne=100&N=4011+20025939
3. http://www.mediastoragecenter.com/scripts/prodlist.asp?idcategory=237&sortField=price&idsft=757&ntype=DE&
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TC-6
Cylinder Savvy: Design 1
Teacher’s Notes in Italics
You are a project designer/engineer hired to make a detailed plan for a CD or DVD
storage container. You will plan three versions of your design and make comparisons
between each one regarding materials needed (surface area) and disc capacity
(volume).
Remind students that a project designer will not have the final word in whether or not
their design gets chosen. Most engineers must present and defend their work to a
panel or board for final decision before too much time and money is spent on
production. So, their final work might be rejected by “the panel” (perhaps other students
will vote or you, their teacher, will decide if their plan is legitimate and functional), but
the goal is to create a design that can be defended. Have them come up with some
goals as they are working. Will their design be for selling at discount stores and
therefore be inexpensive and efficient? Will their design be for selling to the rich and
powerful, and therefore be made to look great, even though storage capacity is limited?
You will need:
•
•
•
•
a metric ruler
a DVD or CD
a mini disc (or metric measurements for one)
sketching paper
Design 1:
1. Sketch a design for holding 12 discs securely. It must be in the shape of a
rectangular prism, triangular prism, cylinder, or other three dimensional figure for
which you can calculate the surface area and volume. Label all dimensions.
Check to see that student’s design is something they can calculate the surface area and
volume for so they will be able to estimate the amount of supplies it will take to create
their actual product.
If their design works, be sure they have labeled all dimensions correctly in the same
units (cm).
2. What do you predict will be your overall surface area and volume in centimeters?
Surface area prediction: ____________ square centimeters
Disc storage volume prediction: ____________ cubic centimeters
Predictions need to be close to accurate, but not exact. In this problem, they are giving
themselves a chance to check for their own understanding of the two concepts. If they
largely over or underestimate, it may mean they are still developing their concept of
surface area and volume. They can look back to this problem later and see if future
estimations/predictions are more accurate since having more practice.
3. Now find the total surface area of Design 1. _______________
Remind students that a company needs surface area information to plan ahead for
buying supplies to create their product – fabric, cardboard, plastics, etc. How much
material will be needed is essential to planning for purchase. Buy too much and
materials go to waste. Buy too little, and you cannot produce as many items.
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4. Find the volume of Design 1. Only calculate the volume for space that would be
used to store discs. _________________
Because students will have non-cylindrical designs, forcing them to calculate cylindrical
volume taken up by discs assures they are getting the same practice with cylinders as
everyone else. You will see on #6 that we are asking students to consider the efficiency
of their design and cylinder (or disc-used space) versus total volume will be considered.
5. Were your predictions close?
_________
Why or why not?
___________________________________________________________
Likely reasons for inaccurate predictions:
a. poor understanding of surface area and/or volume (added or multiplied in
the wrong place(s))- did not address formulas, just looked and calculated
something
b. poor understanding of a cylinder and how to use its parts for calculations –
Area of radius x Height, for example to find area of lateral surface;
squaring diameter to find area of circles
c. incorrect usage of formulas – when students are using complicated
formulas and not being asked to keep track of every little cm, they tend to
write nothing down, and therefore follow order of operations incorrectly
d. incorrect rounding – 23cm rounded to 30cm, for example could throw off
results
Share these likely problems with students and let them change their answer to number
5. Seeing all the ways they can be inaccurate can help them to avoid these mistakes in
the future.
6. Do you have any unused space that is a part of your design, but does not contain
discs? If so, find the volume of the extra, unused space.
________________________________________________________________
What percent of your total volume is for actual disc storage? _______________
Why would this be important to consider? ______________________________
The volume prediction will help them decide if the design is efficient. In other words, if
their design takes $50 in materials to create and only holds 12 discs, it will be less likely
the company will want to take the risk to create them for selling. Someone’s design
costing only $25 that holds 12 discs would likely beat out the more expensive model for
production. What might cause the opposite, however? If your company was designing
high-end models for wealthy consumers, for example, the less efficient design may win
out. It depends on the goals of the company.
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TC-7
Cylinder Savvy: Design 2
Name _______________
The manufacturer wants you to prepare a prototype that is similar in design, but will hold
more discs. Read and complete the following.
1. Sketch a taller version of Design 1, capable of holding 24 discs. Change only
one dimension in your plan, so that you can hold twice as many discs as before.
That is, the outer appearance of your design should only grow in one dimension ,
the height. Label all dimensions.
The goal here is more to double capacity than to double height. If their design
requires a doubled length or width, that is fine. The idea is that their discs will stack
thicker than before. That is, if design looks like a roll of lifesavers on their side, it will
just get longer. If it is a wallet-style design, they may need more sleeves inside to
accommodate more discs. All these designs will yield a greater volume and surface
area, so they will work. However, you may need to highlight with the class, a few
examples which are more standard – a taller tower, for example. This will allow
students to see the way cylinders’ surface area and volume are affected when the
height of a cylinder changes.
1. How do you predict the change in height will affect your overall surface area and
volume in centimeters? Why?
________________________________________________________________
Surface area prediction: ____________ square centimeters
Disc storage volume prediction: ____________ cubic centimeters
Both should increase compared to their original design’s measurements.
2. Now calculate the total surface area of Design 2. _______________
Surface area should grow by adding more height to the lateral surface. In other
words, you are adding a rectangular portion that will wrap around your cylinder to
extend the height. It should not double. If student did not have an original design
that is a cylinder, you will have to check their work for accuracy.
3. Find the volume of the portion in Design 2 which stores discs.
_________________
The volume of a cylinder will double when you have doubled the height. It will be as
though you have just stacked an identical cylinder on top of the original, so now it
will hold twice as much. If students did not have an original design that is a cylinder,
you will have to check their work.
4. What percent of your total volume is for actual disc storage? _______________
Are there any changes in your percentage? Why or why not?
_______________________________________________________________
The percentage should not change unless the students’ new design is not similar
in shape to their original or the arrangement of discs within is adjusted.
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5. Were your predictions correct about your change in surface area and volume?
Have students improved their abilities to make more accurate predictions? Ask
them to compare how far off they were this time with the last time we made
predictions. Although this is a different situation, they may have taken more care in
predicting for this new situation, knowing you would ask them about their accuracy.
6. What conclusions can you draw about the way surface area changed?
_______________________________________________________________
_______________________________________________________________
_______________________________________________________________
See number 3 above.
7. What conclusions can you draw about the way volume changed?
_______________________________________________________________
_______________________________________________________________
_______________________________________________________________
See number 4 above.
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TC-8
Cylinder Savvy: Design 3
Name _______________
Now the manufacturing company has realized there is a market out there for storing
smaller, mini discs. Find the necessary measurements for a mini disc and then
complete the following.
1. Sketch a narrower version of Design 1, for storing mini discs. Make it the same
height as Design 1. (In other words, take your first design and change only the
measurements needed to affect capacity according to radius/diameter.) Label all
dimensions.
MiniDisc’s are 64mm in diameter, as was shown earlier. Their thickness is essentially
the same as a CD/DVD. Again, as before, if design is not cylindrical, check individual
work. Use examples of cylinders that have become narrower to demonstrate to the
class how the surface area and volume will be affected.
2. Sketch each of the 3 surfaces of your original design below. Then shade the
regions that will still be a part of the new, smaller design. Use this to help you
make your predictions.
Sketches should be of two circles, each with a shaded circle within them, and a long
rectangle, with a shorter rectangle shaded inside of it.
3. How do you predict the change in radius/diameter will affect your overall surface
area and volume? Why?
________________________________________________________________
Surface area prediction: ____________ square centimeters
Disc storage volume prediction: ____________ cubic centimeters
The surface area change is complicated. If students draw out the three surfaces and
compare this to their sizes before the smaller diameter was put into effect, they can see
better how things have changed.
4. Find the surface area of Design 3. _________________
The surface area has shrunk by two crescent or doughnut shapes due to reduced radii.
There is also a reduced circumference for those circle bases, so the lateral surfaces
length has shortened, taking away considerable more area. You will have to check
student work for accuracy.
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5. Find the volume of Design 3. _________________
Change in volume is harder to visualize than surface area. Perhaps an enthusiastic
student looking for challenge work could build you a model to demonstrate how the
volume will be impacted- some sort of cylinder imbedded in a larger cylinder.
Answer should not be half the original volume, but should be reduced. Due to the
formula used in finding volume, we can see that changing the radius will not change the
volume by that same amount. Actually, it will be by that amount squared, since the first
operation we perform in the formula is to square the radius. So, if we half the radius,
then our volume should be one half squared, or one forth the original size. This
problem does not use a neat whole number example that is easy to see, so you will
need to discuss with students other possible scenarios to investigate how the volume
changes when radius is changed. Also, see extra practice worksheet, Silo. [10 and T10]
Check student work for accuracy.
6. What percent of your total volume is for actual disc storage? _______________
Are there any changes in your percentage? Why or why not?
_______________________________________________________________
Again, there should be no change in percentage, unless student has altered significantly
the way discs are organized/stored within design.
7. What general conclusions can you make about the way changing height or
diameter affects the surface area of a design?
____________________________________________________________
It is complex, requiring multiple operations to accurately calculate the change.
8. What general conclusions can you make about the way changing height or
diameter affects the volume used up by discs in your design?
____________________________________________________________
Changing only the height of a figure will change the volume directly. That is, double the
height and you double the volume; halve the height and you halve the volume.
Changing the radius/diameter of a cylinder impacts the volume very differently. This is
due to the role of radius in helping us find volume. It will need to be squared before we
multiply it by pi and the height. Therefore, if we double the radius, we will quadruple the
area of the circle and therefore quadruple the volume of our cylinder as well.
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TC-9
Answer Key: Silo
Extension
All answers given using 3.14 for pi.
1. SA = 350π sq.ft. ≈ 1099 sq. ft.
V = 750π cu. Ft. ≈ 2355 cu. ft.
2. SA = 800π sq.ft. ≈ 2512 sq. ft.
V = 3000π cu.ft. ≈ 9420 cu. ft.
3. Increases by 450π sq.ft ≈ 1413 sq. ft.; More than doubled.
Sketches should indicate there is added area to both the circle surfaces (in the
shape of a doughnut or crescent) and the lateral surface (in the form of a rectangle).
4. Volume quadrupled. This is due to the fact that the doubled radius will be
squared, so 25 is used in the original calculations and 100 will be the new radius
squared. Pi and the height, however, will be unchanged.
5. Area of the small circle will be π sq. units or 3.14 sq. units. Area of the large
circle will be 4π sq. units or 12.56 sq. units, 4 times the area of the smaller circle,
even though the radius has only doubled. This is like our silo situation because
the volume varies directly depending on the area of our base. When the radius
of our base doubles, our base area quadruples. This also quadruples our
volume, because all that remains to be done is to multiply by the height of 30
feet.
6. Possibilty A: You could double the height, but this is probably not what the farmer
wants. His silo would likely be too tall to be secure in a wind storm, etc.
Possibility B: A better solution would be to work the problem backwards. If we
want to double the volume, calculate that, plug it in, and solve for a workable
radius. Volume should be 4710 cu. ft., so divide that by pi and 30. Your answer
will be the radius squared value we want. Take the square root to find that a
radius of 7.07 ft. would get you very close to double the volume. That means
your new diameter should be around 14.14 ft.
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TC-10
Candles: Cylinder Change in Height & Radius
Extension
Name _______________________
With the following problem, you will note how a change in height will affect the surface
area and volume of a cylinder.
1. The Wax Candle Company is planning a new line of candles and figuring out how
best to wrap them. One employee wants to sell candles in 2 sizes, both with a
diameter of 3 inches:
small (3in. X 4 in. tall)
large (3in. X 6 in. tall)
Find the amount of wax needed to create each one (volume) and the
approximate amount of tissue to wrap them (surface area). Record your answers
in the table on number 2. Use 3.14 for pi in your calculations.
This problem requires many steps to complete. You may want to remind students of the
formulas for finding volume and surface area of cylinders or allow them to work in
partners to help each other keep the details straight.
2. Wax costs $.10 per cubic inch. Tissue paper costs $.05 per square inch. Find
the cost of making both a small and a large candle in the sizes listed for number
one. Round your answer to their nearest cent.
Small (3 x 4)
Volume
(cu. in.)
28.26
Surface Area
(sq. in.)
51.81
Wax Cost
($.10/cu. in.)
$2.83
Tissue Cost
($.05/sq. in.)
$2.59
Large (3 x 6)
42.39
70.65
$4.24
$3.53
So, to make and wrap one small and one large candle with a diameter of 3 inches it
would cost the sum of the four cost boxes above ( $2.83 + $2.59 + $4.24 + $3.53). One
small and one large candle would cost $13.19 to supply both the wax and tissue..
3. Another employee wants to sell candles that have diameters of 4 inches because
she claims it will save the company money on wax and on tissue for wrapping.
Her designs will be:
small (4in. X 2in. tall)
large (4in. X 4in. tall)
Fill out the table below in order to make comparisons to number 2.
Surface Area
(sq. in.)
50.24
Wax Cost
($.10/cu. in.)
Tissue Cost
($.05/sq. in.)
Small (4 x 2)
Volume
(cu. in.)
25.12
$2.51
$2.51
Large (4 x 4)
50.24
75.36
$5.02
$3.77
Cylinder Savvy
Teacher Materials Page 18 of 19
So, to make and wrap one small and one large candle with a diameter of 4 inches, it
would cost ($2.51 + $2.51 + $5.02 + $3.77). One of each would cost $13.81 to supply.
4. Which size, the 3 or 4 inch diameter will save the Wax Candle Company more
money? Why? Be sure to mention both the affect on surface area and volume.
The three inch diameter will save about $.62 per pair of candles in that size. Even
though at first sight, we might think that the product of the dimensions could key us in to
which design will save money, they are misleading. After careful calculations, we see
that the 3in. diameter will cost less.
5. In your work, you used 3.14 instead of pi to make your calculations. Using pi
would have given a more accurate measurement for surface area and volume.
Why are using 3.14 and rounding your answers to the nearest cent acceptable
for solving this situation? Name a situation for which it would be better to use pi
to find a more accurate answer.
This problem asks us to estimate in advance of planning a purchase. This is an
excellent time to use estimation because the outcome does not demand exactness. If
we were off by a little, no one would lose their life over it; no bridge would fall, no
building would crumble. We may end up spending a bit more or less than we had
planned for, but we still would have chosen the most cost effective method to acquire
our candles for selling.
Cylinder Savvy
Teacher Materials Page 19 of 19