Chapter 12.5, Problem 18HP

To determine

To name: the dimensions of a cube

Answer to Problem 18HP

The dimension of the cube is $\sqrt{2}\text{in}\text{.}$ by $\sqrt{2}\text{in}\text{.}$ by $\sqrt{2}\text{in}\text{.}$ .

Explanation of Solution

Given:

a surface area is between 10 and 20 square inches

Calculation:

The lateral area L of a square prism is, $L=P\cdot h$ .

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

The surface area S of a square prism is, S = L + 2B .

Here, L is the lateral area and B is the area of the base.

The base of the prism is a square whose side is l .

The perimeter P of the base of the prism is, P = 4l .

Here, l is the side of the square base.

Substitute P = 4l and h = lin $L=P\cdot h$ .

Thus,

  $\begin{array}{c}L=P\cdot h\\ =4l\cdot l\\ =4l^2\end{array}$

Therefore, the lateral area of the square prism is $4l^2$ .

Find the surface area of the square prism as follows:

The area B of the base of the prism is, $B=l^2$

Here, l is the side of the square base.

Substitute $B=l^2$ and $L=4l^2$ in S = L + 2B

  $\begin{array}{c}S=L+2B\\ =4l^2+2\cdot l^2\\ =6l^2\end{array}$

Therefore, the surface area of the square prism is $6l^2$ .

The surface area of the cube is between 10 and 20 square inches.

Take the surface area of the cube is 12 square inches.

Thus,

  $\begin{array}{c}6l^2=12\\ l^2=\frac{12}{6}\\ =2\\ l=\sqrt{2}\end{array}$

Conclusion:

Therefore, the dimension of the cube is $\sqrt{2}\text{in}\text{.}$ by $\sqrt{2}\text{in}\text{.}$ by $\sqrt{2}\text{in}\text{.}$ .