Chapter 12.3, Problem 20PPS

i.

To determine

Complete the table for cylinders.

Answer to Problem 20PPS

The completed table is:

Cylinder	Radius	Height	Volume ${\left(yd\right)}^3$
Original	$5$	$3$	$235.5$
Multiply radius by $3$	$5.3=$	$3$	$2119.5$
Multiply height by $3$	$5$	$3$.   $3=$	$706.5$
Multiply both by $3$	$5.3=$	$3.3=$	$6358.5$

Explanation of Solution

Given:

Cylinder	Radius	Height	Volume ${\left(yd\right)}^3$
Original	$5$	$3$	-
Multiply radius by $3$	$5.3=$	$3$	-
Multiply height by $3$	$5$	$3$.   $3=$	-
Multiply both by $3$	-	-	-

Concept Used:

Volume of cylinder is calculated by,

  $V=\pi {\left(r\right)}^2\left(h\right)$

Where,

  $V=$ volume of cylinder.

  $r=$ radius of cylinder.

  $h=$ height of cylinder.

Calculation:

As given in the table,

1. Original:

after putting the given values in formula,

  $V=\pi {\left(r\right)}^2\left(h\right)$

We get:

  $V=\pi {\left(5\right)}^2\left(3\right)$

  $V=235.5y{d}^3$

2. multiply radius by $3$:

after putting the given values in formula,

  $V=\pi {\left(r\right)}^2\left(h\right)$

We get:

  $V=\pi {\left(5.3\right)}^2\left(3\right)$

  $V=2119.5y{d}^3$

3. multiply height by $3$:

after putting the given values in formula,

  $V=\pi {\left(r\right)}^2\left(h\right)$

We get:

  $V=\pi {\left(5\right)}^2\left(3.3\right)$

  $V=706.5y{d}^3$

4. Multiply both by $3$:

after putting the given values in formula,

  $V=\pi {\left(r\right)}^2\left(h\right)$

We get:

  $V=\pi {\left(5.3\right)}^2\left(3.3\right)$

  $V=\pi {\left(15\right)}^2\left(9\right)$

  $V=6358.5y{d}^3$

Cylinder	Radius	Height	Volume ${\left(yd\right)}^3$
Original	$5$	$3$	$235.5$
Multiply radius by $3$	$5.3=$	$3$	$2119.5$
Multiply height by $3$	$5$	$3$.   $3=$	$706.5$
Multiply both by $3$	$5.3=$	$3.3=$	$6358.5$

Conclusion:

Thus, completed table is as above.

ii.

To determine

Compare the original volume to the other volumes.

Answer to Problem 20PPS

So from the above calculation it is observed that original volume of cylinder is smaller than the other one’s.

Explanation of Solution

Given:

Concept Used:

Volume of cylinder is calculated by,

  $V=\pi {\left(r\right)}^2\left(h\right)$

Where,

  $V=$ volume of cylinder.

  $r=$ radius of cylinder.

  $h=$ height of cylinder.

Calculation:

The original volume of the cylinder is calculated as,

Volume of cylinder is calculated by,

  $V=\pi {\left(r\right)}^2\left(h\right)$

  $V=\pi {\left(5\right)}^2\left(3\right)$

  $V=235.5y{d}^3$

While the other volumes calculated in $1^{st}$ part are:

1. multiply radius by $3$:

  $V=2119.5y{d}^3$

2. multiply height by $3$ :

  $V=706.5y{d}^3$

3. Multiply both by $3$:

  $V=6358.5y{d}^3$

So from the above calculation it is observed that original volume of cylinder is smaller than the other one’s.

Conclusion:

The result is as above.

iii.

To determine

Equation to find the volume of cylinder after dilation of $d$

Answer to Problem 20PPS

Required equation is:

  $V=V_{1}ofl\arg ecylinder-V_2\text{}ofSmall\text{}\text{}Cylinder$

Explanation of Solution

Given:

Concept Used:

Volume of cylinder is calculated by,

  $V=\pi {\left(r\right)}^2\left(h\right)$

Where,

  $V=$ volume of cylinder.

  $r=$ radius of cylinder.

  $h=$ height of cylinder.

Calculation:

The given condition is dilation of $d$ ,

So after dilation of $d$ , the equation will be

  $V=Vofl\arg ecylinder-V\text{}ofSmall\text{}\text{}Cylinder$

  $V=\pi {\left(r_1\right)}^2\left(h_1\right)-\pi {\left(r_2\right)}^2\left(h_2\right)$

Where,

Height of the large cylinder $\left(h_2\right)$

Height of the small cylinder $\left(h_1\right)$

Radius of the small cylinder $\left(r_2\right)$

Rradius of the large cylinder $\left(r_1\right)$

Conclusion:

Equation is as mentioned above.