 # A sodium chloride crystal in the shape of a cube is expanding at the rate of 60 cubic microns per second. How fast is the side of the cube growing when the volume is 1000 cubic microns? vangstosiis 2022-07-22 Answered
A sodium chloride crystal in the shape of a cube is expanding at the rate of 60 cubic microns per second. How fast is the side of the cube growing when the volume is 1000 cubic microns?
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The sodium chloride is a cube.
$V={s}^{3}$
Take the derivative with respect to time, t.
$\frac{dV}{dt}=3{s}^{2}\frac{ds}{dt}$
We know that:
$\frac{dV}{dt}=60$
We want to find $\frac{ds}{dt}$ when V=1000, which means that s=10.
$60=3{\left(10\right)}^{2}\frac{ds}{dt}$
$60=300\frac{ds}{dt}$
$\frac{ds}{dt}=\frac{1}{5}$ microns/second
When the cube has a volume of 1000 ${\text{microns}}^{3}$, the side of the cube is growing at a rate of $\frac{1}{5}$ $\text{microns/second}$.