Question

a box with a hinged lid is to be made out of a rectangular piece of cardboard that measures 4 inches by 8 inches. six squares will be cut from the cardboard: one square will be cut from each of the corners, and one square will be cut from the middle of each of the 8 - inch sides. (see figure 1.) the remaining card will be folded to form the box and its lid. (see figure 2.) letting x represent the side - lengths (in inches) of the squares, use the aleks graphing calculator to find the value of x that maximizes the volume enclosed by this box. then give the maximum volume. round your responses to two decimal places.

Explanation:

Step1: Determine box dimensions

Length \(l = 8 - 3x\), width \(w=4 - 2x\), height \(h = x\).

Step2: Form volume function

\(V(x)=(8 - 3x)(4 - 2x)x=24x-28x^{2}+6x^{3}\).

Step3: Find derivative

\(V'(x)=24 - 56x+18x^{2}\).

Step4: Set derivative to 0

\(18x^{2}-56x + 24=0\), solve for \(x\).

Step5: Use graphing calculator

Use ALEKS graphing calculator to find \(x\) value for maximum of \(V(x)\) and corresponding \(V\) value. Round to two - decimal places.

Answer:

Value of \(x\) that maximizes volume: \(0.85\) in
Maximum volume: \(12.32\) in³