## Calculus 8th Edition

$80424.772$ $mm^3/s$
Given information: $\displaystyle \frac{dr}{dt} = 40$ mm/s, $d = 80$mm What we're trying to find: Change in volume with respect to time ($\displaystyle \frac{dV}{dt}$) Diameter = 2$\cdot$radius, d = 2r, therefore $80 = 2r \rightarrow r=40$ mm --- Volume of a sphere: $\displaystyle V = \frac{4}{3}πr^3$ Implicitly differentiate with respect to time: $\displaystyle \frac{dV}{dt} = 4πr^2\cdot\frac{dr}{dt}$ Now we can plug in our givens: $\displaystyle \frac{dV}{dt} = 4π(40)^2(4)$ $\displaystyle \frac{dV}{dt} = 80424.772$ $mm^3/s$