Presentation on theme: "§ 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."— Presentation transcript:
1 § 12.11. 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.
2 Do you think it works for other geometric figures? 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?cabDo you think it works for other geometric figures?
3 It is good to have an easy one once in a while! 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!
4 The ratios are the same for Sphere/Cylinder = 2/3 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 Volume -CylinderArea - 6πr Volume πr 3The ratios are the same for Sphere/Cylinder = 2/3
5 5. Find the volume of a unit regular octagon. Dissect it into two pyramids. The trick is to find the altitude of the pyramid.h 2 = 1 2 – (√2/2) 2 = √2/21hV = (2) (1/3) (1) (√2/2) = √2/3√2
6 V/time = 29,370 cubic feet per hour. 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?481V = bh/3 = 100,017,216 cubic feet.756Time = hoursV/time = 29,370 cubic feet per hour.That is a volume about the size of over two classrooms per hour!!
7 Volume - (1/2)(5000)(π 302) = 7068583 cubic feet 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) = cubic feet/420 = 16,830 dump truck loads
8 8. Investigate the Archimedean solids 8. Investigate the Archimedean solids. What characteristics do they have in common?Faces are regular polygons.
9 V = πhr 2 is fixed. Solve for h h = V/ πr 2 h 6. What is the shape of the cylinder with minimum surface area for a given volume?V = πhr 2 is fixed. Solve for hh = V/ πr 2hrSA = 2πr 2 + 2πrh and substitute for h.SA = 2πr 2 + (2πr)(V/ πr 2)SA = 2πr 2 + 2V/r and take the derivative0 = 4r - 2V/r 2 and take the derivative