1a. There is a regular cylinder, which has a height equal to
its radius. If the radius and height are both increased by 50%, by what % does
the total volume of the cylinder increase?

1b. If the radius and height are both decreased by 10%, by
what % does the total volume of the cylinder decrease?

1c. If the radius is increased by 20% and the height is
decreased by 40%, what % of the volume of the original cylinder does the volume
of the new cylinder represent?

1d. If the radius is increased by 40% and the height is
decreased by 20%, what % of the volume of the original cylinder does the volume
of the new cylinder represent?

1e. If the height is increased by 125%, 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 50%, by what % does
the total surface area of the cube increase?

3a. If the volume of a cube increases by 72.8%, 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 radius
of 5. The first cylinder has a height of 2, the second has a height of 4, the
third a height of 6, etc. The last cylinder has a height of 50. What is the sum
of the volumes of all the cylinders (express your answer in terms of pi)?

1b. Like the previous question: 1

1d. 1.4

Answer key: (Each question should not take you more than 30 seconds to solve if you really understand the concepts involved.)

1a. The volume of a cylinder is πr

^{2}x h (height). The radius itself will be squared and the height stays at constant ratio. The volume will increased thus (1.5)^{3}- 1^{3}-- the original 100% of the volume = 2.375**=237.5%**

1b. Like the previous question: 1

^{3}- 0.9

^{3}[when it's discount/percentage decrease, you use the 100% or 1 - the discount/decrease percentage] = 0.271 =

**27.1% decrease**

1c. 1.2

^{2}[100% + 20% increase = 1.2] x 0.6 [100% -40% = 0.6] = 0.864 or**86.4% of the original volume**

1d. 1.4

^{2}[100% + 40% increase = 1.4] x 0.8 [100% -20% = 0.8] = 1.568 =

**156.8% of the original volume**

1e. When the height of a cylinder is increased 125%, the total volume is is 225% of the original cylinder, or 9/4.

Since the radius is used two times (or squared), it has to decrease 4/9

1 - (2/3) = 1/3^{1/2}= 2/3 for the new cylinder to have the same volume as the old one. [9/4 times 4/9 = 1 or the original volume.]**=**

**0.3**

**= 33.**

**3%**

2. Surface area is 2-D so 1.5

^{2}- 1 = 1.25 =**125% increase**
3a. If a volume of a cube is increased by 72.8 percent, it's 172.8% or 1.728 of the original volume. Now you are going from 3-D (volume) to 2-D (surface area). 1.728

^{2/3}= 1.44 or**44% increase**. [Don't forget to minus 1 (the original volume) since it is asking you the percentage increase.]
3b. From surface area, you can get the side increase by using 1.44

Or you can also use 1.728^{1/2}= 1.2, so 20% increase.^{1/3}= 1.2; 1.2 - 1 =**20%**
4. The volume of a cylinder is πr

^{2}x h . (2 + 4 + 6 + ...50) x 5^{2}π = (25 x 26) x 25π =**16250π**