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

Answer:

The answer in the procedure

Step-by-step 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

Answer 2

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?