Final answer:
The question is about calculating the rate at which the surface area of a cube is increasing given the rate of volume increase using calculus. By differentiating the volume and the surface area with respect to time, and then substituting the given edge length, we can find the required rate.
Step-by-step explanation:
The student is asking about the rate of change of the surface area of a cube given that the volume of the cube is increasing at a constant rate. To find this, we can use the concept of differentials in calculus.
The volume of a cube (V) is related to its edge length (s) by the formula V = s³. The surface area (SA) of a cube is given by SA = 6s². If the volume is increasing at 10 cm³/min, we need to find how fast the surface area is increasing when the edge length is 90 cm.
Step-by-step Explanation:
We start by differentiating the volume with respect to time (t) to get dV/dt, which we know is 10 cm³/min.
Next, we differentiate the volume formula with respect to s to find dv/ds, then solve for ds/dt.
Using the derived value of ds/dt, we then differentiate the surface area formula with respect to s to find d(SA)/ds and multiply it by ds/dt to get d(SA)/dt.
By substituting s = 90 cm into the final expression, we can find the rate at which the surface area increases.
By following these steps, we can conclude that the surface area of the cube is increasing at a certain rate when the edge length is 90 cm.