This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here's the solution to the first rate problem: 1. The radius of a sphere is increasing at 0.5 cm/s. When r = 12 cm, find the rates at which the surface area and the volume are increasing. Given: (dr)/(dt) = 0.5 cm/s r = 12 cm Formulas for a sphere: Surface Area: S = 4 r^2 Volume: V = (4)/(3) r^3 a. Rate of increase of Surface Area ((dS)/(dt)) Step 1: Differentiate the surface area formula with respect to time t. S = 4 r^2 (dS)/(dt) = (d)/(dt)(4 r^2) Using the chain rule, we get: (dS)/(dt) = 4 (2r) (dr)/(dt) (dS)/(dt) = 8 r (dr)/(dt) Step 2: Substitute the given values into the derivative. (dS)/(dt) = 8 (12 cm) (0.5 cm/s) (dS)/(dt) = 8 (6 cm^2/s) (dS)/(dt) = 48 cm2/s b. Rate of increase of Volume ((dV)/(dt)) Step 3: Differentiate the volume formula with respect to time t. V = (4)/(3) r^3 (dV)/(dt) = (d)/(dt)((4)/(3) r^3) Using the chain rule, we get: (dV)/(dt) = (4)/(3) (3r^2) (dr)/(dt) (dV)/(dt) = 4 r^2 (dr)/(dt) Step 4: Substitute the given values into the derivative. (dV)/(dt) = 4 (12 cm)^2 (0.5 cm/s) (dV)/(dt) = 4 (144 cm^2) (0.5 cm/s) (dV)/(dt) = 4 (72 cm^3/s) (dV)/(dt) = 288 cm3/s That's 2 down. 3 left today — send the next one.