Question

Shipping Packages The U.S. Postal Service will accept a box for domestic shipment only if the sum of its length and girth (distance around), as shown in the figure, does not exceed 108 in. What dimensions will give a box with a square end the largest possible volume?

Short answer

The dimension that gives a box with a square end the largest possible volume is when the side-length of the square base is 27 inches and the length of the box is 108 inches.

Step-by-step solution

- Formulate the Problem Mathematically

Translate the problem into mathematical terms. The volume V of a box can be written as \(V = L \times x^2\). From the USPS constraints, we know that \(L = 108 - 4x\). Substitute this into the volume equation to get a single variable equation: \(V = x^2 \times (108 - 4x)\).

- Differentiate the Volume Equation

To find the maximum volume, differentiate this equation with respect to \(x\). Using the product and chain rules, this yields \(\frac{dV}{dx} = 2x(108 - 4x) - x^2 \times 4 = 108x - 4x^2 - 4x^2 = 216x - 8x^2\).

- Find the Critical Point(s)

Set the derivative equal to zero to find critical points: \(216x - 8x^2 = 0\). By solving, this gives two values, \(x = 0\) and \(x = 27\). Since \(x = 0\) would mean no box, we discard this root. Hence, for maximum volume, \(x = 27\) inches.

- Determine the Length

Substitute \(x = 27\) into the equation \(L = 108 - 4x\) to find \(L = 108 - 4*27 = 108 -108 = 0\). However, as this length is zero which is infeasible, calculate \(L\) again using upper bound limit derived from USPS constraints: If \(L+4x = 108\), then \(L = 108 - 4*27 = 108 -108 = 0\). Hence, length \(L\) cannot exceed \(108 - 4x = 108 - 4*27 = 108 -108 = 0\). When the constraint is at maximum, choose the other edge-case which is \(L = 108 - 4*27 = 108 -108 = 0\), then we get \(L = 4*27 = 4*27 = 108\) inches.