Answer
$80424.772$ $mm^3/s$
Work Step by Step
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
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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:
$\displaystyle \frac{dV}{dt} = 4π(40)^2(4)$
$\displaystyle \frac{dV}{dt} = 80424.772$ $mm^3/s$
