Chapter 12.8, Problem 20PPS

To determine

the volumes of the three aquariums.

Answer to Problem 20PPS

The volume of the cylinder $A$ is $139020.69c{m}^3$ .

The volume of the cylinder $B$ is $469194.83c{m}^3$ .

The volume of the cylinder $C$ is $58649.35c{m}^3$ .

Explanation of Solution

Given:

The dimensions of the largest are $1\frac{1}{2}$ the size

The dimensions of the smallest are $\frac{3}{4}$ the size.

  

Formula used:

The volume $V$ of the cylinder is calculated as

  $V=\pi r^2.h$

Calculation:

Consider there are three cylinders. One cylinder is larger than the one in the figure and the other is smaller than the one in the figure.

Consider the dimensions of the largest cylinder are $1\frac{1}{2}$ the size of the above cylinder. Also the dimensions of the smallest are $\frac{3}{4}$ the size of the one shown.

First convert the mixed number into rational number.

  $\begin{array}{l}1\frac{1}{2}=\frac{2+1}{2}\\ =\frac{3}{2}\end{array}$

The objective is to find the volume of the three cylinders.

Let $A$ represents are cylinder in the figure, $B$ represents the larger cylinder and $C$ represents the smaller cylinder.

First find the volume of the cylinder $A$ .

The volume $V$ of the cylinder is

  $V=\pi r^2.h$

Where $r$ is the radius and is $h$ is the height of the cylinder.

Since radius is always half that of diameter, so the radius of the cylinder is $\frac{76.2}{2}cm$ that is $38.1cm$

The volume $V_1$ of the cylinder $A$ is

  $\begin{array}{l}{V}_1=\pi r^{2}h\\ =\pi {\left(38.1\right)}^2\left(30.5\right)\\ =\pi \left(1451.61\right)\left(30.5\right)\\ =\pi \left(44274.105\right)\\ \approx \left(3.14\right)\left(44274.105\right)[Since\pi \approx 3.14]\\ \approx 139020.69\end{array}$

Therefore, the volume of the cylinder $A$ is approximately $139020.69c{m}^3$ .

Two solid figures are similar if they have the same shape and their corresponding linear measurements are proportional.

Since both the figures are cylinder and their corresponding linear measurements are $1\frac{1}{2}$ the size of the other, so the corresponding linear measurements are proportional. Hence the cylinders are similar.

The ratio of the diameter of cylinder $A$ to cylinder $B$ is $\frac{2}{3}$

The scale factor is $\frac{2}{3}$ .

Now, use the property that if two solids are similar with scale factor of $\frac{a}{b}$ then the volume is in the ratio ${\left(\frac{a}{b}\right)}^3$ .

Thus,

  $\frac{VolumeofcylinderA}{Volumeofcylinder\text{ B}}={\left(\frac{a}{b}\right)}^3$ , where $\frac{a}{b}$ is the scale factor

  $\begin{array}{l}\frac{\text{139020}\text{.69}}{\text{Volume of cylinder B}}\text{=}{\left(\frac{\text{2}}{\text{3}}\right)}^{\text{3}}\text{=}\frac{\text{8}}{\text{27}}\\ \left(\text{139020}\text{.69}\right)\text{27=8×Volume of cylinder B}\\ \text{8×Volume of cylinder B=}\left(\text{139020}\text{.69}\right)\text{27}\\ \text{ =3753558}\text{.63}\\ \text{Volume of cylinder B=}\frac{\text{3753558}\text{.63}}{\text{8}}\\ \text{ »469194}\text{.83}\end{array}$

Therefore, the volume of the cylinder $B$ is $469194.83c{m}^3$

Next find the volume of the cylinder $C$ .

Since both the figures are cylinder and their corresponding linear measurements are $\frac{3}{4}$ the size of the other, so the corresponding linear measurements are proportional. Hence the cylinders are similar.

The ratio of the diameter of cylinder $A$ to cylinder $C$ is $\frac{4}{3}$ .

The scale factoris $\frac{4}{3}$ .

Now, use the property that if two solids are similar with scale factor of $\frac{a}{b}$ then the volume is in the ratio ${\left(\frac{a}{b}\right)}^3$ .

Thus,

  $\frac{VolumeofcylinderA}{Volumeofcylinder\text{ C}}={\left(\frac{a}{b}\right)}^3$ , where $\frac{a}{b}$ is the scale factor

  $\begin{array}{l}\frac{\text{139020}\text{.69}}{\text{Volume of cylinder B}}\text{=}{\left(\frac{\text{4}}{\text{3}}\right)}^{\text{3}}\text{=}\frac{\text{64}}{\text{27}}\\ \left(\text{139020}\text{.69}\right)\text{27=8×Volume of cylinder C}\\ \text{64×Volume of cylinder C=}\left(\text{139020}\text{.69}\right)\text{27}\\ \text{ =3753558}\text{.63}\\ \text{Volume of cylinder C=}\frac{\text{3753558}\text{.63}}{\text{64}}\\ \text{ »58649}\text{.35}\end{array}$

Therefore, the volume of the cylinder $C$ is $58649.35c{m}^3$

Conclusion:

Therefore, the volume of the cylinder $A$ is $139020.69c{m}^3$ .

The volume of the cylinder $B$ is $469194.83c{m}^3$ .

The volume of the cylinder $C$ is $58649.35c{m}^3$ .