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Draw several kinds of triangles including a right triangle. Draw a square on each of
the sides of the triangles. Compute the areas of the squares and use this information
to investigate whether the Pythagorean Theorem works for only right triangles. Use
a geometry utility if available.
Solution:
The Pythagorean formula is a special case of a more general equation.
The full equation is the Cosine Law:
C2 = A2 + B2 - 2AB cos(c)
This equation only degenerates into the Pythagorean formula when cos(c) is equal to zero and
that happen when angle c is 90 degrees.
If the angle is not 90 degrees, you end up with something where the two squares don't add up
to the third
For example, 3 squared plus 4 squared equals 5 squared, and the sides of the triangle are 3,
4, and 5 when the angle is 90 degrees, but if you make the angle 77o then the side lengths
become 4, 4, and 5. And 4 squared plus 4 squared does not equal 5 squared.
So Pythagorean Theorem works for only right triangles
Find the surface area of each of the following:
(a)
Surface Area of a Cube = 6 a 2= 6*(4)2=96 sq. cm
(b)
Surface Area of a Cylinder = 2 pi r 2 + 2 pi r h
= pi (6)2 +2 pi *6*12
=678.584013 sq.cm
(c)
Surface Area of a Rectangular Prism = 2ab + 2bc + 2ac
= 2(5)(6)+2(6)(8)+2(5)(8)
=236 sq.cm.
(d)
Surface area of sphere = 4r2 = 4 *3.14 *(4)2=201.06193 square cm
(e)


Surface area of Rt circular cone = r  rl
=  (6) +  (6) (8)
=169.646003 sq. cm
(f)
It has four triangles and one base.
Area of a triangle = 1/2 base * height.
=1/2 * 5.0 * 6.5 = 16.25 sq. cm. for each triangle.
So for four triangles = 16.25 * 4 = 65 sq. cm. (1)
One base = 5 * 5 = 25 sq. cm.
-(2)
Adding (1) and (2) we get
65 + 25 = 90 square centimeters
(g)
Total surface area = (2
= 2
r2) + (2  r h)
(10)2 + 2  (10)(60)= 4 398.22972 sq. ft.

3. Jeremy has a fish tank that has a 40-cm by 70-cm rectangular base. The water is 25 cm
deep. When he drops rocks into the tank, the water goes up by 2 cm. What is the volume in
liters of the rocks?
Solution:
volume of the rock
40 x 70 x 25 = 70,000 (no rocks)
40 x 70 x 27 = 75,600 (with the rocks)
Volume of rocks = difference in volumes=5600 cubic centimeters
Since 1 milliliter = 1 cubic centimeter
5600 milliliters
volume of rocks in liters. =5.6 liters
18. The Great Pyramid of Cheops has a square base of 771 ft on a side and a height of 486
ft. How many apartments 35 ft × 20 ft ×8 ft would be needed to have a volume equivalent
to that of the Great Pyramid’s?
Solution:
V (pyramid) = (1/3) A * h = (1/3) (771)2 * 486 =96 299 442
V (apartment) = 35 * 20 * 8 = 5600
V (pyramid)/V (apartment) = 96 299 442/5600 = 17 196.3289 cu. Ft.
21. A heavy metal sphere with radius 10 cm is dropped into a right circular cylinder with base radius
of 10 cm. If the original cylinder has water in it that is 20 cm high, how high is the water after the
sphere is placed in it?
Solution::
Answering
25. A tennis ball can in the shape of a cylinder holds three tennis balls snugly. If the radius of
a tennis ball is 3.5 cm, what percentage of the tennis ball can is occupied by air?
Solution:
percent occupied by air = (total volume - volume of balls)/total volume
= 1 - volume of balls/total volume
= 1 - 3 * (4/3)  r3/(  r2 * 6r)
= 1 - 4  r3/(6  r3)
= 1 - 4/6
= 1/3=33.3%
34. A right cylindrical can is to hold approximately 1 L of water. What should be the
height of the can if the radius is 12 cm?
Solution: The volume for a right circular cylinder is
V  r 2 h
1 L = 1000 cm3
So,
The volume of the cylinder is given by:
So the height can be solved for:
52. Explain how you would find the volume of an irregular shape.
Solution: measuring the volume of irregular shaped figure:
Partly fill the measuring cylinder with water and take an initial reading: level 1 in the
figure
2. Now drop the object into the water so that it is completely covered and read water
level 2
The volume of the object is the difference between the two water level readings.
1.