91Ó°ÊÓ

Substitute $2p^3$ for r and $4p^3$ for h into $V=\mathrm{Ï€}{r}^{2}h$.

$V=\mathrm{Ï€}{\left(2p^3\right)}^2\left(4p^3\right)$

Apply the property of power of product.

$V=\mathrm{Ï€}4p^6â‹\dots 4p^3$

Apply property of product of powers to the above obtained expression.

$$\begin{gathered} V=16\mathrm{Ï€}{p}^{6+3} \\ =16\mathrm{Ï€}{p}^9 \end{gathered}$$

Therefore, the volume of cylinder is$16\mathrm{Ï€}{p}^9$ cubic units.

The product of the square of the coefficient of radius and coefficient of height must be 16. The exponents of the radius and the height must have a sum of 9.

The possible values of radius are $4p,4p^2,2p^3,2p^4$and $2p$.

The possible values of height are$p^7,p^5,4p^3,4p$ and $4p^7$.

Make a table of possible values of radius and height.

If the height is doubled, $h=2\left(4p^3\right)=8p^3$.

Substitute$2p^3$ for r and$8p^3$ for h into $V=\mathrm{Ï€}{r}^{2}h$.

$V=\mathrm{Ï€}{\left(2p^3\right)}^2\left(8p^3\right)$

Apply the property of power of product.

$V=\mathrm{Ï€}4p^6â‹\dots 8p^3$

Apply property of product of powers to the above obtained expression.

$$\begin{gathered} V=32\mathrm{Ï€}{p}^{6+3} \\ =32\mathrm{Ï€}{p}^9 \end{gathered}$$

Therefore, if the height is doubled then the volume of cylinder becomes$32\mathrm{Ï€}{p}^9$ cubic units.