The surface area of a cube is changing at the rate of 8 cm^2/s. How fast is the volume changing when the surface area is 60cm?

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- May 11th 2010, 11:58 PMscubasteve94Related rates of change
The surface area of a cube is changing at the rate of 8 cm^2/s. How fast is the volume changing when the surface area is 60cm?

- May 12th 2010, 03:41 AMArchie Meade
Hi scubasteve94,

if it is a regular cube, then

$\displaystyle surface\ area=6s^2$

$\displaystyle volume=s^3$

where s=length of all edges

$\displaystyle SA=60\ cm^2=6s^2\ \Rightarrow\ s^2=10\ cm^2\ \Rightarrow\ s=\sqrt{10}\ cm$

$\displaystyle \frac{d}{dt}(SA)=8\ cm^2/sec\ =\ \frac{ds}{dt}\ \frac{d}{ds}6s^2=\frac{ds}{dt}12s=\frac{ds}{dt}12\ sqrt{10}\ cm$

hence

$\displaystyle \frac{ds}{dt}=\frac{8}{12\sqrt{10}}=\frac{2}{3\sqr t{10}}\ cm/sec$..... which is the rate of change of the length of an edge of the cube.

$\displaystyle \frac{dV}{dt}=\frac{dV}{ds}\ \frac{ds}{dt}=3s^2\ \frac{2}{3\sqrt{10}}=30\ \frac{2}{3\sqrt{10}}=\frac{20}{\sqrt{10}}\ cm^3/sec$