Final Answer:
A rectangular prism with dimensions x, y, and z is being filled with water at a rate of 5 cubic units per minute. The dimensions are changing, and at a particular moment, x = 3 units, y = 4 units, and z = 5 units. Determine the rates at which each side is changing with respect to time.
Step-by-step explanation:
Consider a rectangular prism where x, y, and z are the length, width, and height, respectively. The rate of change of volume (V) with respect to time (t) is given as dV/dt = 5 cubic units per minute. At a specific moment when x = 3, y = 4, and z = 5, we need to find the rates at which each side is changing (dx/dt, dy/dt, dz/dt).
The volume of the rectangular prism is given by V = xyz. Applying the product rule and substituting the given values, we have dV/dt = yz(dx/dt) + xz(dy/dt) + xy(dz/dt). Solving for dx/dt, dy/dt, and dz/dt using the provided values, we can determine how each side is changing with respect to time.
This exercise illustrates a real-world application of related rates, where understanding the changing dimensions of a rectangular prism as it is filled with water allows us to calculate the rates at which each side is changing. The application involves the application of the product rule and substitution to derive the rates of change for each dimension.