Q. 44 - Calculus

Question

For Eliza, Alina wants to make a rectangular box whose base is twice as long as it is wide. This box will be lined on the entire inside with velvet and in addition, the outside of the top of the box is to be lined in velvet. If Alina has $240$ square inches of velvet, how can she make Eliza’s box so that it holds as many keepsakes as possible?

Step-by-Step Solution

Answer

Ans:  The dimensions of the box are $4.47\times 8.94\times 5.96$ and the volume is approximately $238.4$ square inches.

Step 1. Given information.

given, 

    The total available velvet for the box is $240$ square inches.

Step 2. The objective is to find the dimensions of the largest box that can be made using the velvet.

Let, a box of length $2w$ inches, width $w$ inches, and height $h$inches. 

The surface area of the box is,

       $\begin{array}{r}S=2wh+2\left(2w^2\right)+2\left(2wh\right)\\ =6wh+4w^2\end{array}$

Consider the surface area of the box is $240$ square inches.

Therefore, 

      $\begin{array}{r}6wh+4w^2=240\\ 3wh+2w^2=120\end{array}$

Step 3. The volume of the box is,

  $\begin{array}{r}V=\left(2w\right)\left(w\right)\left(h\right)\\ =2w^{2}h\end{array}$

Substitute the value of $wh$ from the surface area equation,

     $\begin{array}{r}V\left(w\right)=2w\left(\frac{120-2w^2}{3}\right)\\ =80w-\frac{4}{3}{w}^3\end{array}$

Step 4. Find the value of w and v as follows:

Differentiate the function with respect to $w$,

   $V^{\mathrm{\prime }}\left(w\right)=80-4w^2$

Equate the derivate to zero and solve for $w$,

    $\begin{array}{r}80-4w^2=0\\ w^2=20\\ w\approx 4.47\end{array}$

The value of $n$ is,

   $\begin{array}{r}h=\frac{120-2w^2}{3w}\\ =\frac{120-2\left(20\right)}{3\left(4.47\right)}\\ \approx 5.96\end{array}$

The volume of the box is,

   $\begin{array}{r}V=2\left(20\right)\left(5.96\right)\\ \approx 238.4\end{array}$

Therefore, the dimensions of the box are $4.47\times 8.94\times 5.96$ and the volume is approximately $238.4$ square inches.