Problem 20 - Calculus of a Single Variable

Question

Maximum Volume A rectangular solid (with a square base) has a surface area of 337.5 square centimeters. Find the dimensions that will result in a solid with maximum volume.

Step-by-Step Solution

Answer

The dimensions of the base square are \( x \approx 7.5 \) cm, and the height of the rectangular solid is \( y \approx 15 \) cm. These dimensions will result in a solid with maximum volume.

Step 1: Formulation of The Equations

Let's denote the sides of the square base as x (in cm) and the height of the rectangular solid as y (in cm). The surface area (S) and the volume (V) of the solid are given by the formulas: \( S = 2x^2 + 4xy \) and \( V = x^2y \). Given that S = 337.5 cm^2, we substitute into the surface area equation, which results in: \( 2x^2 + 4xy = 337.5 \) or simplified, \( y = (337.5-2x^2)/(4x) \).

Step 2: Substitute y in Volume Equation

Substitute y from step 1 into the volume equation: \( V = x^2((337.5-2x^2)/(4x)) \) which simplifies to \( V = x(84.375 - 0.5x^2) \).

Step 3: Find the Maximum Volume

The volume V will be maximum where the derivative \(V'\) is zero. So, we find the derivative of V:\( V' = 84.375 - 1.5x^2 \). Setting this equal to zero and solving for x, we find \( x = \sqrt{56.25} \), then we substitute x back into \( y = (337.5-2x^2)/(4x) \) to find y.

Step 4: Verifying Maximum Volume

We need to ensure that these dimensions do indeed maximize the volume. For that, we take the derivative of \(V'\), which is \( V'' = -3x \). Substituting the number we obtained for x here, we see that \( V'' < 0 \), which confirms that the volume is maximized.