19. balloons a spherical balloon is inflated and its volume increases at a rate of 15 in³/min. what is the…

Explanation:

Step1: Recall the volume formula for a sphere

The volume formula for a sphere is (V=\frac{4}{3}\pi r^{3}).

Step2: Differentiate the volume formula with respect to time (t)

Using the chain - rule, (\frac{dV}{dt}=\frac{dV}{dr}\cdot\frac{dr}{dt}). First, find (\frac{dV}{dr}): If (V = \frac{4}{3}\pi r^{3}), then (\frac{dV}{dr}=4\pi r^{2}). So (\frac{dV}{dt}=4\pi r^{2}\cdot\frac{dr}{dt}).

Step3: Solve for (\frac{dr}{dt})

We know that (\frac{dV}{dt} = 15\space in^{3}/min) and (r = 10\space in). Substitute the values into the equation (\frac{dV}{dt}=4\pi r^{2}\cdot\frac{dr}{dt}): (15=4\pi(10)^{2}\cdot\frac{dr}{dt}). (15 = 400\pi\cdot\frac{dr}{dt}). Then (\frac{dr}{dt}=\frac{15}{400\pi}=\frac{3}{80\pi}\space in/min).

Answer:

(\frac{3}{80\pi}\space in/min)