Chapter 12.6, Problem 20IP

(a)

To determine

To complete: The table.

Answer to Problem 20IP

The complete table is given below.

Explanation of Solution

Given information:

The given table is shown below.

Cylinder	Radius	Height	Volume $\left({\text{yd}}^3\right)$
Original	5	3	_
Multiply radius by 3	$5\cdot 3=\_$	3	_
Multiply height by 3	5	$3\cdot 3=\_$	_
Multiply both by 3	_	_	_

Calculation:

Write the expression to calculate the volume of the cylinder.

  $V=\pi r^{2}h$...............(1)

Where,

• $r$ is the radius.
• $h$ is the height.

Complete the table using the formula given in equation (1).

Cylinder	Radius	Height	Volume $\left({\text{yd}}^3\right)$
Original	5	3	$\pi {\left(5\right)}^{2}3=75\pi$
Multiply radius by 3	$5\cdot 3=15$	3	$\pi {\left(15\right)}^{2}3=675\pi$
Multiply height by 3	5	$3\cdot 3=9$	$\pi {\left(5\right)}^{2}9=225\pi$
Multiply both by 3	$5\cdot 3=15$	$3\cdot 3=9$	$\pi {\left(15\right)}^{2}9=2025\pi$

Conclusion:

Therefore, the complete table is shown above.

(b)

To determine

To compare: The volume of the original cylinder with the other cylinders.

Answer to Problem 20IP

The comparison is given below.

Explanation of Solution

Given information:

From part (a), the volume of the cylinder is given below.

Cylinder	Radius	Height	Volume $\left({\text{yd}}^3\right)$
Original	5	3	$\pi {\left(5\right)}^{2}3=75\pi$
Multiply radius by 3	$5\cdot 3=15$	3	$\pi {\left(15\right)}^{2}3=675\pi$
Multiply height by 3	5	$3\cdot 3=9$	$\pi {\left(5\right)}^{2}9=225\pi$
Multiply both by 3	$5\cdot 3=15$	$3\cdot 3=9$	$\pi {\left(15\right)}^{2}9=2025\pi$

Calculation:

From the table, the comparison of the other cylinders with the original cylinders is as follows.

When the radius of the cylinder is multiplied by 3 and keep the height same, the volume of the cylinder is 9 times the volume of the original cylinder.

When the height of the cylinder is multiplied by 3 and keep the radius same, the volume of the cylinder is 3 times the volume of the original cylinder.

When 3 multiply both the height and radius of the cylinder, the volume of the cylinder is 27 times the volume of the original cylinder.

Conclusion:

Therefore, the comparison is given above.

(c)

To determine

To write: The equation of a cylinder after a dilation $d$ .

Answer to Problem 20IP

The equation of a cylinder after the dilation $d$ is $V=\pi {\left(r-d\right)}^{2}h$ .

Explanation of Solution

Given information:

The dilation of the cylinder is $d$ .

Calculation:

Write the expression to calculate the volume of the cylinder.

  $V=\pi r^{2}h$...............(1)

Where,

• $r$ is the radius.
• $h$ is the height.

Let the dilation of the cylinder is $d$ . The dilation refers to the stretch for the shrinks. The diameter of the cylinder decreases.

Write the equation for the volume of the cylinder after the dilation.

  $V=\pi {\left(r-d\right)}^{2}h$

Conclusion:

Therefore, the equation of a cylinder after the dilation $d$ is $V=\pi {\left(r-d\right)}^{2}h$ .