Question

A wooden beam has a rectangular cross section of height h and width w. The strength s of the beam is directly proportional to its width and the square of its height. Find the dimensions of the cross section of such a beam of maximum strength that can be cut from a round log of diameter 30 in. Hint: s = kh²w, where k is the constant of proportionality. Round your answers to two decimal places.

Answer 1

The maximum strength beam that can be cut from a round log of 30-inch diameter is a square beam with sides of 15 inches.

Step-by-step explanation:

To find the dimensions of the cross-section of a wooden beam of maximum strength that can be cut from a round log of diameter 30 inches, we need to maximize the strength s, given by the formula s = kh2w, where h is the height, w is the width, and k is the constant of proportionality. Since the log is round, the width and the height are limited by the diameter of the log. We can demonstrate that the maximum strength occurs when the rectangle inscribed in the semicircle (half the log's cross-section) is a square, as this uses the maximum area and therefore the maximum height h.

To inscribe a rectangle with sides of equal length (a square) in a semicircle of diameter d, the side length of the square, which is both the width and the height of our beam, is equal to d/2. So, the maximum dimensions of the beam are 15 inches by 15 inches (since 30 inches is the diameter of the log). The strength of the beam can be calculated with the constant k, if known, by substituting these dimensions into the formula.