## Friday, May 4, 2012

### Dimensional Change Questions II

Dimensional change questions II:   Answer key below.

1a. There is a regular cylinder, which has a height equal to its radius. If the radius and height are both increased by 20%, by what % does the total volume of the cylinder increase?

1b. If the radius and height are both decreased by 20%, by what % does the total volume of the cylinder decrease?

1c. If the radius is increased by 50% and the height is decreased by 25%, what % of the volume of the original cylinder does the volume of the new cylinder represent?

1d. If the radius is increased by 25% and the height is decreased by 50%, what % of the volume of the original cylinder does the volume of the new cylinder represent?

1e. If the height is increased by 300%, what % does the radius need to be decreased by for the volume to remain the same?

2. If the side of a cube is increased by 30%, by what % does the total surface area of the cube increase? By what % does the volume increase?

3a. If the volume of a cube increases by 174.4%, by what % does the total surface area of the cube increase?

3b. By what % did the side length of the cube increase?

4. You have a collection of cylinders, all having a height of 4. The first cylinder has a radius of, 1 the second has a radius of 4, the third a radius of 9,…etc. The last cylinder has a radius of 121 (all square numbers, starting with 12. What is the sum of the volumes of all the cylinders (express your answer in terms of π)?

Answer key to dimensional change questions II:

1a. There is a regular cylinder, which has a height equal to its radius. If the radius and height are both increased by 20%, by what % does the total volume of the cylinder increase?

72.8%

1b. If the radius and height are both decreased by 20%, by what % does the total volume of the cylinder decrease?

48.8% (Only 0.83 = 0.512 = 51.2% of the original percentage left and 100% - 51.2% = 48.8%.)

1c. If the radius is increased by 50% and the height is decreased by 25%, what % of the volume of the original cylinder does the volume of the new cylinder represent?

168.75%

1d. If the radius is increased by 25% and the height is decreased by 50%, what % of the volume of the original cylinder does the volume of the new cylinder represent?

78.125%

1e. If the height is increased by 300%, what % does the radius need to be decreased by for the volume to remain the same?

50%

2. If the side of a cube is increased by 30%, by what % does the total surface area of the cube increase? By what % does the volume increase?

The surface area will increase 69% and the volume will increase 119.7%

3a. If the volume of a cube increases by 174.4%, by what % does the total surface area of the cube increase?

96%

3b. By what % did the side length of the cube increase?

40%

4. You have a collection of cylinders, all having a height of 4. The first cylinder has a radius of, 1 the second has a radius of 4, the third a radius of 9,…etc. The last cylinder has a radius of 121 (all square numbers, starting with 12. What is the sum of the volumes of all the cylinders (express your answer in terms of π)?

The volume of a cylinder is πr2 and the sum of the first n square numbers is n(n+1)(2n+1) over 6.
The answer is 4 x [ 11 (11 +1) (2 x 11 + 1)/6 times π = 2024π.