Q. 60 - Calculus

Question

Suppose you want to make an open-topped box out of a 4×6 index card by cutting a square out of each corner and then folding up the edges, as shown in the figure. How large a square should you cut out of each corner in order to maximize the volume of the resulting box?

Step-by-Step Solution

Answer

The area of the square is $x^2=2 .$

Step 1.Given Information

The car has 6 inches length and 4 inches width.

The dimensions of the card are $x\times x$ .

Step 2.The volume

The volume of the box is:$$\begin{gathered} v=(6\times 4\times x)-4\left(x^3\right) \\ v=24x-4x^3 . \end{gathered}$$

Step 3.The derivative

The derivative is:$$\begin{gathered} v=24x-4x^3 \\ {v}^{\text{'}}=24-12x^2=0 \\ 24=12x^2 \\ x=\sqrt{2} . \end{gathered}$$

The second derivative is calculated as:$$\begin{gathered} v^{\text{'}\text{'}}=-24x<0 \\ =-24\left(\sqrt{2}\right)<0 \end{gathered}$$

Step 4.The solution

The area of the square is $x^2=2 .$