1. prove the pythagorean theorem by a method not used in class
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§ 12.1. 1. Prove the Pythagorean Theorem by a method not used in class. There are over 260 of them. You should not have had too much trouble finding another one. - PowerPoint PPT PresentationTRANSCRIPT
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1. Prove the Pythagorean Theorem by a method not used in class.. § 12.1
There are over 260 of them. You should not have had too much trouble finding another one.
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2. On the three sides of a right triangle construct semicircles with centers at the midpoints of the sides. Calculate the area of each of the three semicircles. Do you see a relationship?
Do you think it works for other geometric figures?
a
b
c2 2
a
1 a aA
2 2 8
2 2
b
1 b bA
2 2 8
2 2
c
1 c cA
2 2 8
2 2 2SO a b c8
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3. Find the ratio of the volume to the surface area of a cube.
It is good to have an easy one once in a while!
V 1
A 6
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4. A sphere is circumscribed by a cylinder. Find the ratio of the two surface areas. Find the ratio of the two volumes.
Use unit radius.
Sphere –
Area – 4πr 2 Volume -
Cylinder
Area - 6πr 2 Volume - 2 πr 3
The ratios are the same for Sphere/Cylinder = 2/3
3r3
4
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5. Find the volume of a unit regular octagon.
Dissect it into two pyramids. The trick is to find the altitude of the pyramid.
1
√2
h 2 = 1 2 – (√2/2) 2 = √2/2
h
V = (2) (1/3) (1) (√2/2) = √2/3
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6. What is the volume of the Great Pyramid of Giza that had a side measure of 756 ft and an altitude of 481 feet? If it took 30 years of 6 day weeks working 10 hours a day, how many cubic feet were put in place each hour?
V = bh/3 = 100,017,216 cubic feet.
756
481
Time = 93600 hours
V/time = 29,370 cubic feet per hour.
That is a volume about the size of over two classrooms per hour!!
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7. An auto tunnel through a mountain is being planned. It will be semicircular cylinder with a radius of 30 feet and a length of 5000 feet. How many cubic feet of dirt will have to be removed? If a dump truck has a bed of dimensions 7 feet by 10 feed by 6 feet, how many loads will be required to carry away the dirt?
Volume - (1/2)(5000)(π 302) = 7068583 cubic feet
7068583/420 = 16,830 dump truck loads
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8. Investigate the Archimedean solids. What characteristics do they have in common?
Faces are regular polygons.
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6. What is the shape of the cylinder with minimum surface area for a given volume?
V = πhr 2 is fixed. Solve for h
h = V/ πr 2
r
h
SA = 2πr 2 + 2πrh and substitute for h.
SA = 2πr 2 + (2πr)(V/ πr 2)
SA = 2πr 2 + 2V/r and take the derivative
0 = 4r - 2V/r 2 and take the derivative