Question

A rectangular gift box has dimensions that can be represented as shown in the figure. The volume of the box is 672 cubic inches. Which of the following is not a dimension of the box?

A 6 in.

B 7 in.

C 8 in.

D 12 in.

Short answer

6 in. is not a dimension of the box.

Step-by-step solution

Step 1. Write the formula of volume of rectangular box.

Let l, band h be the length, width and height of the rectangular gift box respectively.

Volume of a rectangular box is

$V=l\times b\times h$

Step 2. Apply formula of volume of rectangular box.

Replace V with 672, w with l, b with$w-5$ and h with $w-4$.

$$\begin{gathered} w\left(w-5\right)\left(w-4\right)=672 \\ {w}^3-4w^2-5w^2+20w=672 \\ {w}^3-9w^2+20w-672=0 \end{gathered}$$

Step 3. Find the dimensions of the rectangular box.

By solving the above equation, $w=12$.

Replace w by 12 in $b=w-5$.

$$\begin{gathered} b=w-5 \\ =12-5 \\ =7\text{in} \end{gathered}$$

Replace w by 12 in $h=w-4$.

$$\begin{gathered} h=w-4 \\ =12-4 \\ =8\text{in} \end{gathered}$$

The dimensions of the rectangular box are 12 in, 7 in and 8 in. 6 in is not the dimension of the rectangular box. Therefore, option A is the required answer.