Step 1 :We are given a problem of related rates in calculus. The radius of a sphere is increasing at a constant rate of 2 cm/sec. We need to find how fast the surface area and the volume are increasing when the radius is 10 cm.
Step 2 :The formula for the surface area of a sphere is \(A = 4\pi r^2\) and the formula for the volume of a sphere is \(V = \frac{4}{3}\pi r^3\). We can differentiate these with respect to time to find the rates of change of the surface area and volume with respect to time.
Step 3 :Given that \(r = r\) and \(\frac{dr}{dt} = 2\), we can calculate \(\frac{dA}{dt} = 16\pi r\) and \(\frac{dV}{dt} = 8.0\pi r^2\).
Step 4 :Substituting \(r = 10\) into the above equations, we find that \(\frac{dA}{dt} = 160\pi\) and \(\frac{dV}{dt} = 800.0\pi\).
Step 5 :Final Answer: The rate of change of the surface area when the radius is 10 cm is \(\boxed{160\pi}\) cm²/sec and the rate of change of the volume when the radius is 10 cm is \(\boxed{800\pi}\) cm³/sec.