Two More Problems

1: If the diameter of a right cylindrical can with circular bases is increased by 25%, by what percent should the height be increased in order to double the volume of the original can?

For convenience sake...let the original diameter of the can  =2

So....the original radius  = 1

So...the original volume is

V = pi * (1)^2 * original height

V = pi * original height

original height = V /pi

When the original diameter is increased by 25%, so is the original radius...so the new radius is 1.25

And we have that

2V  = pi * (1.25)^2 * new height

[2V ] / pi * (1.25)^2  =  new height

So

new height

____________   =

original height

[2V] / [ pi *(1.25)^2 ]

_________________      =

   V /pi

      2V                           pi

__________    *     _________

pi * (1.25)^2                V

2 / (1.25)^2  =   

1.28

So....the original height should be increased by [ 1.28  - 1]  =  .28  = 28%    to double the volume

2: A steel sphere with a 3-inch radius is made by removing metal from the corners of a cube that has the shortest possible side lengths. How many cubic inches are in the volume of the cube?

The sphere will be tangent to each face of the cube.....so....the side length of the cube must  be  6 in

So....the volume of the cube  is   just

side^3   =   6^3   =  216 in^3