Final answer:
The rate of change of the volume of the rectangular solid is -30 cubic meters per second.
Step-by-step explanation:
To find the rate of change of the volume of the rectangular solid, we need to first express the volume in terms of its dimensions. The volume of a rectangular solid is given by V = length x width x height. Since the length is constant at 5 meters, we can rewrite the volume as V = 5(width)(height). We are given that the height is increasing at a rate of 2 meters per second and the width is decreasing at a rate of 3 meters per second. So, when the height is 4 meters and the width is 2 meters, the volume can be calculated as V = 5(2)(4) = 40 cubic meters.
To find the rate of change of the volume, we need to differentiate the volume function with respect to time. Since both the height and width are changing with respect to time, we use the product rule of differentiation. Differentiating V with respect to time, we have dV/dt = 5(dwidth/dt)(dheight/dt). Substituting the given rates of change, we have dV/dt = 5(-3)(2) = -30 cubic meters per second.