Chapter 12.8, Problem 21PPS

(a)

To determine

To find:the ratio of the surface areas and the ratio of the volumes.

Answer to Problem 21PPS

The ratio of surface area of prism $A$ $4:9$ and the ratio of volume of prism $B$ $8:27$

Explanation of Solution

Given: Use the two similar prisms at the right.

  

Calculation:

Consider the height of the rectangular prism $A$ is $4ft$ and the surface area of the prism $A$ is $40f{t}^2$ . Also the height of the rectangular prism $B$ is $6ft$ and the surface area of the prism $A$ is $40f{t}^2$ . Consider the two prism are similar.

The objective is to find the ratio of the surface areas and the ratio of the volumes of the two figures.

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 proportional, so the cylinders are similar.

The ratio of the length of $A$ to $B$ is $\frac{4}{6}$ that is $\frac{2}{3}$ .

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

Use the property that if two solids are similar with scale factor of $\frac{a}{b}$ then the surface area is in the ratio ${\left(\frac{a}{b}\right)}^2$ .

Thus,

  $\frac{\text{Surface area of the prism A}}{\text{Surface area of the prism B}}\text{=}{\left(\frac{\text{a}}{\text{b}}\right)}^{\text{2}}$

Where $\frac{a}{b}$ is the scale factor.

  $\begin{array}{l}\frac{\text{Surface area of the prism A}}{\text{Surface area of the prism B}}\text{=}{\left(\frac{\text{2}}{\text{3}}\right)}^{\text{2}}\\ =\frac{2^2}{3^2}\\ =\frac{4}{9}\end{array}$

Therefore, the ratio of the surface are is $4:9$

Now, use the property that if two solids are similar with scale factor of $\frac{a}{b}$ then the volume is in the ratio ${\left(\frac{a}{b}\right)}^2$ .

Thus,

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

  $\begin{array}{l}\frac{VolumeoftheprismA}{Volumeoftheprism\text{ B}}={\left(\frac{2}{3}\right)}^3\\ =\frac{2^3}{3^3}\\ =\frac{8}{27}\end{array}$

Therefore, the ratio of the volume is $8:27$

Conclusion:

Therefore, the ratio of surface area of prism $A$ $4:9$ and the ratio of volume of prism $B$ $8:27$

(b)

To determine

To find:the surface area of prism $B$ .

Answer to Problem 21PPS

The surface area of the rectangular prism $B$ is $90f{t}^2$ .

Explanation of Solution

Calculation:

The objective is to find the surface area of the rectangular prism $B$ .

Put surface of the prism $A$ equal to $40$ in the ratio of the surface area.

  $\frac{SurfaceareaoftheprismA}{Surfaceareaoftheprism\text{ B}}={\left(\frac{a}{b}\right)}^2$ , where $\frac{a}{b}$ is the scale factor.

  $\frac{40}{Surfaceareaoftheprism\text{ B}}=\frac{4}{9}$

  $\begin{array}{l}\text{40}\left(\text{9}\right)\text{=4}\left(\text{SA B}\right)\\ \text{360=4}\left(\text{SA B}\right)\\ \text{4}\left(\text{SA B}\right)\text{=360}\\ \left(\text{SA B}\right)\text{=}\frac{\text{360}}{\text{4}}\\ \text{=90}\end{array}$

Therefore, the surface area of the rectangular prism $B$ is $90f{t}^2$ .

Conclusion:

Therefore, the surface area of the rectangular prism $B$ is $90f{t}^2$ .

(c)

To determine

To find:the volume of prism $A$ .

Answer to Problem 21PPS

The volume of the rectangular prism $A$ is $16f{t}^2$ .

Explanation of Solution

Calculation:

The objective is to find the volume of the rectangular prism $A$ .

Put volume of the prism $B$ equal to $54$ in the ratio of the volumes.

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

  $\begin{array}{l}\frac{VolumeoftheprismA}{54}=\frac{8}{27}\\ VolumeoftheprismA=\frac{54.8}{27}\\ =2.8\\ =16\end{array}$

Therefore, the volume of the rectangular prism $A$ is $16f{t}^2$ .

Conclusion:

Therefore, the volume of the rectangular prism $A$ is $16f{t}^2$ .