Using <span>r
</span> to represent the radius and <span>t
</span> for time, you can write the first rate as:
<span><span><span><span>dr</span><span>dt</span></span>=4<span>mms</span></span>
</span>
or
<span><span>r=r<span>(t)</span>=4t</span>
</span>
The formula for a solid sphere's volume is:
<span><span>V=V<span>(r)</span>=<span>43</span>π<span>r3</span></span>
</span>
When you take the derivative of both sides with respect to time...
<span><span><span><span>dV</span><span>dt</span></span>=<span>43</span>π<span>(3<span>r2</span>)</span><span>(<span><span>dr</span><span>dt</span></span>)</span></span>
</span>
...remember the Chain Rule for implicit differentiation. The general format for this is:
<span><span><span><span><span>dV<span>(r)</span></span><span>dt</span></span>=<span><span>dV<span>(r)</span></span><span>dr<span>(t)</span></span></span>⋅<span><span>dr<span>(t)</span></span><span>dt</span></span></span>
</span>with <span><span>V=V<span>(r)</span></span>
</span> and <span><span>r=r<span>(t)</span></span>
</span>.</span>
So, when you take the derivative of the volume, it is with respect to its variable <span>r
</span> <span><span>(<span><span>dV<span>(r)</span></span><span>dr<span>(t)</span></span></span>)</span>
</span>, but we want to do it with respect to <span>t
</span> <span><span>(<span><span>dV<span>(r)</span></span><span>dt</span></span>)</span>
</span>. Since <span><span>r=r<span>(t)</span></span>
</span> and <span><span>r<span>(t)</span></span>
</span> is implicitly a function of <span>t
</span>, to make the equality work, you have to multiply by the derivative of the function <span><span>r<span>(t)</span></span>
</span> with respect to <span>t
</span> <span><span>(<span><span>dr<span>(t)</span></span><span>dt</span></span>)</span>
</span>as well. That way, you're taking a derivative along a chain of functions, so to speak (<span><span>V→r→t</span>
</span>).
Now what you can do is simply plug in what <span>r
</span> is (note you were given diameter) and what <span><span><span>dr</span><span>dt</span></span>
</span> is, because <span><span><span>dV</span><span>dt</span></span>
</span> describes the rate of change of the volume over time, of a sphere.
<span><span><span><span><span>dV</span><span>dt</span></span>=<span>43</span>π<span>(3<span><span>(20mm)</span>2</span>)</span><span>(4<span>mms</span>)</span></span>
</span><span><span>=6400π<span><span>mm3</span>s</span></span>
</span></span>
Since time just increases, and the radius increases as a function of time, and the volume increases as a function of a constant times the radius cubed, the volume increases faster than the radius increases, so we can't just say the two rates are the same.
