Chapter 12.5, Problem 21HP

To determine

To sketch: two prism such that one has a greater volume and the other has a great surface area.

Answer to Problem 21HP

The figure I has greater volume than figure II but figure II has greater surface area than figure I.

Explanation of Solution

Given:

Consider the figures.

  

Calculation:

The given figure is a rectangular prism.

The objective is to find the volume and surface area of the two figures.

The volume V of a square prism is given by

  $V={\left(side\right)}^3$

The volume V of a rectangular prism is given by

  $V=l\cdot w\cdot h$

Where l is the length, w is the width and w is the height of the rectangular prism.

The lateral area L of a square prism is given by the formula

  $L=P\cdot h$

Where P is the perimeter of the base and h is the height the square prism.

The surface area S of a square prism is given by the formula

S = L + 2B

Where L is the lateral area and B is the area of the base.

Take the square prism whose side is 6 units.

The volume V of a square prism is

  $\begin{array}{l}V={\left(side\right)}^3\\ =6^3\\ =216\end{array}$

Therefore, the volume of the square prism is $216{\text{units}}^3$ .

Take the rectangular prism whose length is 10 units, width is 7 units and height is 3 units.

The volume V of a rectangular prism is

  $\begin{array}{c}V=l\cdot w\cdot h\\ =10\cdot 7\cdot 3\\ =210\end{array}$

Therefore, the volume of the rectangular prism is $210{\text{units}}^3$ .

Next find the surface area of the square prism.

The perimeter P of the base of the prism is given by

  $\begin{array}{c}P=4\left(side\right)\\ =4\left(6\right)\\ =24\end{array}$

Put P = 24 and h = 6 in the lateral area formula to find the lateral area of the prism.

Thus,

  $\begin{array}{c}L=P\cdot h\\ =24\cdot 6\\ =144\end{array}$

Therefore, the lateral area of the square prism is $144{\text{units}}^2$ .

First find the area B of the base of the prism. The area B of the base is given by $B={\left(side\right)}^2$ where l is the length and w is the width of the square base.

  $\begin{array}{c}=6^2\\ =36\end{array}$

Put B = 36 and L = 144 in the surface area formula to find the surface area of the prism.

  $\begin{array}{c}S=L+2B\\ =144+2\cdot 36\\ =144+72\\ =216\end{array}$

Therefore, the surface area of the square prism is $216{\text{units}}^2$ .

Now, find the surface area of the rectangular prism.

The perimeter P of the base of the prism is given by

  $P=2\left(l+w\right)$ Wherel is the length and w is the width of the rectangular base.

  $\begin{array}{c}=2\left(10+7\right)\\ =2\cdot 17\\ =34\end{array}$

Put P = 34 and h = 3 in the lateral area formula to find the lateral area of the prism.

Thus,

  $\begin{array}{c}L=P\cdot h\\ =34\cdot 3\\ =102\end{array}$

Therefore, the lateral area of the rectangular prism is $102{\text{units}}^2$ .

First find the area B of the base of the prism. The area B of the base is given by $B=l\cdot w$ where l is the length and w is the width of the rectangular base.

  $\begin{array}{c}=10\cdot 7\\ =70\end{array}$

Put B = 70 and L = 102 in the surface area formula to find the surface area of the prism.

  $\begin{array}{c}S=L+B\\ =102+2\cdot 70\\ =102+140\\ =242\end{array}$

Therefore, the surface area of the rectangular prism is $242{\text{units}}^2$ .

Conclusion:

Therefore, the figure I has greater volume than figure II but figure II has greater surface area than figure I.