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
Hey Theamazingdivine, good to see you again. Step 1: Identify the given information and the formula. We are given the rate of change of volume (dV)/(dt) = 10 cm^3/min and the current radius r = 5 cm. We need to find the rate of change of the radius (dr)/(dt). The formula for the volume of a sphere is V = (4)/(3) r^3. Step 2: Differentiate the volume formula with respect to time t. Using the chain rule, we differentiate both sides of the volume equation with respect to t: (dV)/(dt) = (d)/(dt)((4)/(3) r^3) (dV)/(dt) = (4)/(3) · 3r^2 (dr)/(dt) (dV)/(dt) = 4 r^2 (dr)/(dt) Step 3: Substitute the given values into the differentiated equation. We have (dV)/(dt) = 10 and r = 5. 10 = 4 (5)^2 (dr)/(dt) 10 = 4 (25) (dr)/(dt) 10 = 100 (dr)/(dt) Step 4: Solve for (dr)/(dt). (dr)/(dt) = (10)/(100) (dr)/(dt) = (1)/(10) cm/min The rate at which the radius is increasing when the radius is 5 cm is (1)/(10) cm/min. (1)/(10) cm/min Send me the next one 📸