Chapter 11.1, Problem 42PPE

(a)

To determine

Find the ratio of surface area to the volume for each figure.

Answer to Problem 42PPE

Ratio ${\text{SA}}_{\text{Square}\text{Prism}}:{\text{V}}_{\text{Square}\text{Prism}}=2(b+2h):bh$ and ${\text{SA}}_{\text{Cylinder}}:{\text{V}}_{\text{Cylinder}}=2(r+h):r·h$

Explanation of Solution

Given:

To keep heating costs for a building, architects want the ratio of surface area to volume to be as small as possible. What is an expression for the ratio of Surface area to volume of each figure?

The figures:

  

Concept Used:

Surface area of square prism is: ${\text{SA}}_{\text{Square}\text{Prism}}=2b^2+4bh$

Volume of square prism: ${\text{V}}_{\text{Square}\text{Prism}}=b^2·h$

Surface area of Cylinder: ${\text{SA}}_{\text{Cylinder}}=2·\text{π}·r^2+2·\text{π}·r·h$

Volume of cylinder: ${\text{V}}_{\text{Cylinder}}=\text{π}·r^2·h$

Calculation:

For square prism:

  $\frac{{\text{SA}}_{\text{Square}\text{Prism}}}{{\text{V}}_{\text{Square}\text{Prism}}}=\frac{2b^2+4·b·h}{b^2·h}=\frac{2·b·(b+2h)}{b·b·h}=\frac{2(b+2h)}{b·h}$

Ratio: ${\text{SA}}_{\text{Square}\text{Prism}}:{\text{V}}_{\text{Square}\text{Prism}}=2(b+2h):bh$

For Cylinder:

  $\frac{{\text{SA}}_{\text{Cylinder}}}{{\text{V}}_{\text{Cylinder}}}=\frac{2·\text{π}·r^2+2·\text{π}·r·h}{\text{π}·r^2·h}=\frac{2·\text{π}·r(r+h)}{\text{π}·r·r·h}=\frac{2(r+h)}{r·h}$

  ${\text{SA}}_{\text{Cylinder}}:{\text{V}}_{\text{Cylinder}}=2(r+h):r·h$

Thus, ratio ${\text{SA}}_{\text{Square}\text{Prism}}:{\text{V}}_{\text{Square}\text{Prism}}=2(b+2h):bh$ and ${\text{SA}}_{\text{Cylinder}}:{\text{V}}_{\text{Cylinder}}=2(r+h):r·h$

(b)

To determine

For each figure, find the ratio of surface area to volume when b = 12 ft, h = 18 ft, and r = 6 ft.

Answer to Problem 42PPE

  ${\text{SA}}_{\text{Square}\text{Prism}}:{\text{V}}_{\text{Square}\text{Prism}}=4:9$ and ${\text{SA}}_{\text{Cylinder}}:{\text{V}}_{\text{Cylinder}}=4:9$

Explanation of Solution

Given:

To keep heating costs for a building, architects want the ratio of surface area to volume to be as small as possible. What is an expression for the ratio of Surface area to volume of each figure?

The figure:

  

For each figure, what is the ratio of surface area to volume when b = 12 ft, h = 18 ft, and r = 6 ft.

Concept Used:

Ratio ${\text{SA}}_{\text{Square}\text{Prism}}:{\text{V}}_{\text{Square}\text{Prism}}=2(b+2h):bh$ and ${\text{SA}}_{\text{Cylinder}}:{\text{V}}_{\text{Cylinder}}=2(r+h):r·h$

Calculation:

Ratio ${\text{SA}}_{\text{Square}\text{Prism}}:{\text{V}}_{\text{Square}\text{Prism}}=2(b+2h):bh$ and ${\text{SA}}_{\text{Cylinder}}:{\text{V}}_{\text{Cylinder}}=2(r+h):r·h$

Find the ratio of surface area to volume when b = 12 ft, h = 18 ft, and r = 6 ft.

  $\begin{array}{l}b=12\text{ft};h=18\text{ft};r=6\text{ft};\text{find}{\text{SA}}_{\text{Square}\text{Prism}}:{\text{V}}_{\text{Square}\text{Prism}}\\ {\text{SA}}_{\text{Square}\text{Prism}}:{\text{V}}_{\text{Square}\text{Prism}}=2(b+2h):bh=2(12+2·18):12·18=96:216=4:9\end{array}$

  ${\text{SA}}_{\text{Square}\text{Prism}}:{\text{V}}_{\text{Square}\text{Prism}}=4:9$

  $\begin{array}{l}b=12\text{ft};h=18\text{ft};r=6\text{ft};\text{find}{\text{SA}}_{\text{Cylinder}}:{\text{V}}_{\text{Cylinder}}\\ {\text{SA}}_{\text{Cylinder}}:{\text{V}}_{\text{Cylinder}}=2(r+h):r·h=2(6+18):6·18=48:108=4:9\end{array}$

  ${\text{SA}}_{\text{Cylinder}}:{\text{V}}_{\text{Cylinder}}=4:9$

Thus, ${\text{SA}}_{\text{Square}\text{Prism}}:{\text{V}}_{\text{Square}\text{Prism}}=4:9$ and ${\text{SA}}_{\text{Cylinder}}:{\text{V}}_{\text{Cylinder}}=4:9$