Chapter 12.8, Problem 16IP

(a)

To determine

To find: The surface area of the box.

Answer to Problem 16IP

The surface area of the box is $432{\text{in}}^2$ .

Explanation of Solution

Given Information:

The volume of the rectangular prism-shaped shipping box is 576 cubic inches. The length of the box is 12 inches and the width is 8 inches.

Calculation:

Write the expression to calculate the volume of the shipping box.

  $\text{Volume}=lwh$

Where,

• $l$ is the length of the box.
• $w$ is the width of the box.
• $h$ is the height of the box.

Substitute the values in the above expression.

  $\begin{array}{c}\text{576}=12\times 8\times h\\ h=\frac{576}{12\times 8}\\ =6\text{inches}\end{array}$

Write the expression to calculate the surface area of the box.

  $\text{Surface}\text{area}=2\left(lw+wh+hl\right)$

Substitute the values in the above expression.

  $\begin{array}{c}\text{Surface}\text{area}=2\left(12\cdot 8+8\cdot 6+6\cdot 12\right)\\ =2\left(96+48+72\right)\\ =432{\text{in}}^2\end{array}$

Conclusion:

Therefore, the surface area of the box is $432{\text{in}}^2$ .

(b)

To determine

To find: The box with greater surface area.

Answer to Problem 16IP

The box with length, width, and height of another shipping box is 16 inches, 9 inches, and 4 inches has greater surface area.

Explanation of Solution

Given Information:

From part (a), the original shipping box has surface area of 432 square inches. The length, width, and height of another shipping box is 16 inches, 9 inches, and 4 inches, respectively.

Calculation:

Write the expression to calculate the surface area of another shipping box.

  $\text{Surface}\text{area}=2\left(lw+wh+hl\right)$

Substitute the values in the above expression.

  $\begin{array}{c}\text{Surface}\text{area}=2\left(16\cdot 9+9\cdot 4+4\cdot 16\right)\\ =2\left(144+36+64\right)\\ =488{\text{in}}^2\end{array}$

From the above, it is clear that the surface area of another box is greater than the surface area of the original box.

Conclusion:

Therefore, the box with length, width, and height of another shipping box is 16 inches, 9 inches, and 4 inches has greater surface area.

(c)

To determine

To predict: The shape of a rectangular prism that has the same volume but different surface area than either of the boxes.

Answer to Problem 16IP

A rectangular prism with length 24 inches, width 4 inches, and height 6 inches has the same volume but different surface area than either of the boxes.

Explanation of Solution

Given Information:

From part (a), the volume of the original shipping box is 576 cubic inches and surface area is 432 square inches.

Calculation:

Let’s consider a rectangular prism with length 24 inches, width 4 inches, and height 6 inches.

Write the expression to calculate the volume of the shipping box.

  $\text{Volume}=lwh$

Where,

• $l$ is the length of the box.
• $w$ is the width of the box.
• $h$ is the height of the box.

Substitute the values in the above expression.

  $\begin{array}{c}\text{Volume}=24\cdot 4\cdot 6\\ =576{\text{in}}^3\end{array}$

Write the expression to calculate the surface area of another shipping box.

  $\text{Surface}\text{area}=2\left(lw+wh+hl\right)$

Substitute the values in the above expression.

  $\begin{array}{c}\text{Surface}\text{area}=2\left(24\cdot 4+4\cdot 6+6\cdot 24\right)\\ =2\left(96+24+144\right)\\ =528{\text{in}}^2\end{array}$

From the above, it is clear that the volume is similar in both the case but the surface area is different.

Conclusion:

Therefore, the rectangular prism with length 24 inches, width 4 inches, and height 6 inches has the same volume but different surface area than either of the boxes.