Question

The dimensions in inches of a shipping box at we ship $4$ You can be expressed as width $x$, length $x+5$, and height $3x–1$. The volume is about $7.6{\mathrm{ft}}^3$. Find the dimensions of the box in inches. Round to the nearest inch.

Answer 1

Step-1: Find the value of $x$:

The formula for the volume is $\mathrm{volume}=\left(\mathrm{lenght}\times \mathrm{width}\times \mathrm{height}\right)$.

Convert the unit of volume from feet to inches:

$\begin{array}{rcl}1{\mathrm{foot}}^3& =& 1728{\mathrm{inches}}^3\\ & \Rightarrow & 7.6{\mathrm{feet}}^3=7.6\times 1728{\mathrm{inches}}^3\\ & \Rightarrow & 7.6{\mathrm{feet}}^3=13132.8{\mathrm{inches}}^3\end{array}$

Substitute $\mathrm{width}=x\mathrm{inches},\mathrm{length}=\left(x+5\right)\mathrm{inches},\mathrm{height}=\left(3x-1\right)\mathrm{inches}$ in the volume formula $\mathrm{volume}=\left(\mathrm{lenght}\times \mathrm{width}\times \mathrm{height}\right)$:

$\begin{array}{rcl}\mathrm{volume}& =& \mathrm{length}\times \mathrm{width}\times \mathrm{height}\\ & \Rightarrow & 13132.8=x\times \left(x+5\right)\times \left(3x-1\right)\\ & \Rightarrow & 13132.8=3x^3+14x^2-5x\\ & \Rightarrow & x\approx 15\mathrm{in}.\end{array}$

Thus, the width of the box is $15\mathrm{in}$.

Step-2: Find the length of the box:

Substitute $x=15$ in $x+5$ to find length:

$\begin{array}{rcl}x+5& =& 15+5\\ & =& 20\mathrm{in}.\end{array}$

Thus, the length of the box is $20\mathrm{in}$.

Ste-3: Find the height of the box:

Substitute $x=15$ in $3x–1$ to find height:

$\begin{array}{rcl}3x-1& =& 3\left(15\right)-1\\ & =& 44\mathrm{in}.\end{array}$

Thus, the length of the box is $44\mathrm{in}.$

Therefore, the required dimensions of the box will be $\mathbf{20}\mathbf{}\mathbf{in}\mathbf{\times }\mathbf{15}\mathbf{}\mathbf{in}\mathbf{\times }\mathbf{}\mathbf{44}\mathbf{}\mathbf{in}$.