Q61. - Algebra 1

Question

PACKAGING For a commercial art class, Aiko must design a new container for individually wrapped pieces of candy. The shape that she chose is a cylinder. The formula for the volume of a cylinder is $V=\pi r^{2}h$.

a. The radius that Aiko would like to use is $\mathbf{2}{p}^3$, and the height is $\mathbf{4}{p}^3$. Write a monomial that represents the volume of her container.

b. Make a table of values for five possible radius widths and heights if the volume is to remain the same.

c. What is the volume of Aiko’s container if the height is doubled?

Step-by-Step Solution

Answer

a. The volume of cylinder is $\mathbf{16}\pi p^9$ cubic units.

b. Table of possible values of radius and heights is

 

Radius	Height
$\mathbf{4}p$	$p^7$
$\mathbf{4}{p}^2$	$p^5$
$\mathbf{2}{p}^3$	$\mathbf{4}{p}^3$
$\mathbf{2}{p}^4$	4p
$\mathbf{2}p$	$\mathbf{4}{p}^7$

 

c. If the height is doubled then the volume of cylinder becomes $\mathbf{32}\pi p^9$ cubic units.

Part a. Step 1. Substitution.

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

$V=\pi {\left(2p^3\right)}^2\left(4p^3\right)$

Part a. Step 2. Power of product.

Apply the property of power of product.

$V=\pi 4p^6\cdot 4p^3$

Part a. Step 3. Product of powers.

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

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

Therefore, the volume of cylinder is $16\pi p^9$ cubic units.

Part b. Step 1. Choose values.

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.

Part b. Step 2. Possible values of radius and height.

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 $4p^7$.

Part b. Step 3. Make a table.

Make a table of possible values of radius and height.

 

Radius	Height
$4p$	$p^7$
$4p^2$	$p^5$
$2p^3$	$4p^3$
$2p^4$	$4p$
$2p$	$4p^7$

 

Part c. Step 1. Substitution.

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

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

$V=\pi {\left(2p^3\right)}^2\left(8p^3\right)$

Part c. Step 2. Power of product.

Apply the property of power of product.

$V=\pi 4p^6\cdot 8p^3$

Part c. Step 3. Product of powers.

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

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

Therefore, if the height is doubled then the volume of cylinder becomes $32\pi p^9$ cubic units.