§ 12.1
This presentation is the property of its rightful owner.
Sponsored Links
1 / 9

1. Prove the Pythagorean Theorem by a method not used in class.. PowerPoint PPT Presentation


  • 30 Views
  • Uploaded on
  • Presentation posted in: General

§ 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.

Download Presentation

1. Prove the Pythagorean Theorem by a method not used in class..

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


1 prove the pythagorean theorem by a method not used in class

§ 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.


1 prove the pythagorean theorem by a method not used in class

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?

c

a

b

Do you think it works for other geometric figures?


1 prove the pythagorean theorem by a method not used in class

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!


1 prove the pythagorean theorem by a method not used in class

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


1 prove the pythagorean theorem by a method not used in class

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 = √2/2

1

h

V = (2) (1/3) (1) (√2/2) = √2/3

√2


1 prove the pythagorean theorem by a method not used in class

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?

481

V = bh/3 = 100,017,216 cubic feet.

756

Time = 93600 hours

V/time = 29,370 cubic feet per hour.

That is a volume about the size of over two classrooms per hour!!


1 prove the pythagorean theorem by a method not used in class

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


1 prove the pythagorean theorem by a method not used in class

8. Investigate the Archimedean solids. What characteristics do they have in common?

Faces are regular polygons.


1 prove the pythagorean theorem by a method not used in class

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

h

r

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


  • Login