7. the volume v of a sphere of radius r changes over time t.\na. find an equation relating dv/dt to…

Explanation:

Step1: Volume formula of sphere

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

Step2: Differentiate (V) with respect to (t)

Using the chain - rule (\frac{dV}{dt}=\frac{dV}{dr}\cdot\frac{dr}{dt}). Differentiate (V = \frac{4}{3}\pi r^{3}) with respect to (r): (\frac{dV}{dr}=4\pi r^{2}). So, (\frac{dV}{dt}=4\pi r^{2}\frac{dr}{dt}).

Step3: Solve part b

Given (\frac{dr}{dt} = 2) in/min and (r = 4) inches. Substitute into (\frac{dV}{dt}=4\pi r^{2}\frac{dr}{dt}). (\frac{dV}{dt}=4\pi(4)^{2}\times2) (=4\pi\times16\times 2) (=128\pi\approx 402.12) in³/min.

Step4: Solve part c

Given (\frac{dV}{dt}=10) in³/min and (r = 5) inches. From (\frac{dV}{dt}=4\pi r^{2}\frac{dr}{dt}), we can solve for (\frac{dr}{dt}). (\frac{dr}{dt}=\frac{\frac{dV}{dt}}{4\pi r^{2}}) Substitute (\frac{dV}{dt}=10) and (r = 5) (\frac{dr}{dt}=\frac{10}{4\pi(5)^{2}}=\frac{10}{100\pi}=\frac{1}{10\pi}\approx0.0318) in/min.

Answer:

a. (\frac{dV}{dt}=4\pi r^{2}\frac{dr}{dt}) b. (\frac{dV}{dt}\approx402.12) in³/min c. (\frac{dr}{dt}\approx0.0318) in/min