Chapter 7.1, Problem 61PPS

a.

To determine

To write: A monomial that represents the volume of her cylinder.

Answer to Problem 61PPS

The required volume is $16\pi p^9$

Explanation of Solution

Given information:

The shape of the container Alka chose is a cylinder.

The formula of the volume of a cylinder is $V=\pi r^{2}h$

The radius that Alka would like to use is $2p^3$

The height that Alka would like to use is $4p^3$

Calculation:

  $V=\pi r^{2}h$

(Write the formula)

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

(Substitute the values of r and h)

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

(Power of a product)

  $=\pi 2^2{p}^{6}4p^3$

(Power of a power)

  $=\pi \left(2^2\right)4p^6{p}^3$

(Commutative)

  $=\pi \left(2^2\right)4p^9$

(Product of powers)

  $=16\pi p^9$

(Simplify)

Thus the volume is $16\pi p^9$ .

b.

To determine

To make: A table for five possible measures for the radius and height of a cylinder having the same volume.

Answer to Problem 61PPS

The required table is:

Radius	Height	Volume
$2p^4$	$4p$	$16\pi p^9$
$2p$	$4p^7$	$16\pi p^9$
$2p^2$	$4p^5$	$16\pi p^9$
$4p^3$	$p^3$	$16\pi p^9$
$p^3$	$16p^3$	$16\pi p^9$

Explanation of Solution

Given information:

The shape of the container Alka chose is a cylinder.

The formula of the volume of a cylinder is $V=\pi r^{2}h$

The radius that Alka would like to use is $2p^3$

The height that Alka would like to use is $4p^3$

Calculation:

  $V=\pi r^{2}h$

The original volume is $16\pi p^9$

Radius	Height	Volume
$2p^4$	$4p$	$16\pi p^9$
$2p$	$4p^7$	$16\pi p^9$
$2p^2$	$4p^5$	$16\pi p^9$
$4p^3$	$p^3$	$16\pi p^9$
$p^3$	$16p^3$	$16\pi p^9$

c.

To determine

To find: The volume of Alka’s container if the height is doubled.

Answer to Problem 61PPS

The required answer is $32\pi p^9$ .

Explanation of Solution

Given information:

The shape of the container Alka chose is a cylinder.

The formula of the volume of a cylinder is $V=\pi r^{2}h$

The radius that Alka would like to use is $2p^3$

The height that Alka would like to use is $4p^3$

The height of the cylinder is doubled.

Calculation:

The new height is $2\cdot 4p^3$

  $=8p^3$

  $V=\pi r^{2}h$

(Write the formula)

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

(Substitute the values of r and h)

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

(Power of a product)

  $=\pi {\left(2\right)}^2{p}^6\left(8p^3\right)$

(Power of a power)

  $=\pi {\left(2\right)}^2\cdot 8\cdot p^6{p}^3$

(Commutative)

  $=\pi \cdot 4\cdot 8\cdot p^9$

(Product of powers)

  $=32\pi p^9$

(Simplify)

Thus the new volume is $32\pi p^9$