The resistance of a beam with a rectangular cross section is proportional to the product of its width and the square of its height. Find the dimensions of the strongest rectangular beam that can be extracted from an elliptical cylindrical trunk with cross sections of major axis 24cm and minor axis 16cm.

According to the problem, the resistance of the beam is proportional to the product of its width and the square of its height. Therefore, we can express the resistance (R) as: R = k * w * h^2 Where "k" is a constant of proportionality. Now, we need to maximize the product of width and the square of height (w * h^2). To do this, we must consider that the rectangular beam must fit within the elliptical cross-section of the trunk, meaning that: 2 * h = 16 cm (since the minor axis of the ellipse is twice the height of the rectangular beam) h = 8 cm Now, we need to find the corresponding width (w) that maximizes the product w * h^2 within the constraints of the elliptical trunk. The major axis of the ellipse is 24 cm, which is twice the width of the rectangular beam (2 * w). 2 * w = 24 cm w = 12 cm So, the width of the rectangular beam is 12 cm, and the height is 8 cm. These are the dimensions of the strongest rectangular beam that can be extracted from the elliptical cylindrical trunk.