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.
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Here's the solution to the next rate problem: 2. Gas is escaping from a spherical balloon at 45 m^3/min. How fast is the radius decreasing when r = 5 m? Given: Rate of change of volume, (dV)/(dt) = -45 m^3/min (negative because the volume is decreasing) Radius, r = 5 m The formula for the volume of a sphere is V = (4)/(3) r^3. Step 1: Differentiate the volume formula with respect to time t. V = (4)/(3) r^3 Using the chain rule, we differentiate both sides 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 2: Substitute the given values into the derivative and solve for (dr)/(dt). -45 m^3/min = 4 (5 m)^2 (dr)/(dt) -45 = 4 (25) (dr)/(dt) -45 = 100 (dr)/(dt) (dr)/(dt) = (-45)/(100) (dr)/(dt) = (-9)/(20) m/min The negative sign indicates that the radius is decreasing. The rate at which the radius is decreasing is (9)/(20) m/min. 3 done, 2 left today. You're making progress.