Question

The standard formula for the volume of a cylinder is V = πr2h. If the cylinder is scaled proportionally by a factor of k, its volume becomes V' = V × k3. Use your algebra skills to derive the steps that lead from V = πr2h to V' = V × k3 for a scaled cylinder. Show your work.

Answer 1

So, r' (the new r) becomes kr, and h'=h*k,

so V' = pi * (r')^2*h' = pi * (r*k)^2*(h*k) = pi*r^2*h*k^3 = V * k^3

Convinced?

Answer 2

The answer in the procedure

Explanation:

we know that

If two figures are similar, then the ratio of its corresponding sides is equal to the scale factor

so

`k=r'/r` and
`k=h'/h`

The volume of the original cylinder is equal to

`V=\pi r^(2)h`

If the cylinder is scaled proportionally by a factor of k

then

the new radius is ------>
`r'=kr`

the new height is ------>
`h'=kh`

The volume of the scaled cylinder is equal to

`V'=\pi r'^(2)h'`

substitute the values

`V'=\pi (kr)^(2)(kh)`

`V'=(k^(3))\pi r^(2)h`

Remember that

`V=\pi r^(2)h`

so

substitute

`V'=V(k^(3))`

The volume of the scaled cylinder is equal to the scale factor elevated to the cube multiplied by the volume of the original cylinder