Question

the radius ( r ) of a sphere is increasing at a rate of 4 inches per minute.
(a) find the rate of change of the volume (in ( mathrm{in}^{3} / mathrm{min} )) when ( r = 11 ) inches and when ( r = 34 ) inches.
( r = 11 ) in ( mathrm{in}^{3} / mathrm{min} )
( r = 34 ) in ( mathrm{in}^{3} / mathrm{min} )
(b) explain why the rate of change of the volume of the sphere is not constant even though ( \frac{d r}{d t} ) is constant.
( \bigcirc ) if ( \frac{d r}{d t} ) is constant, ( \frac{d v}{d r} ) is equal to ( t^{2} ).
( \bigcirc ) if ( \frac{d r}{d t} ) is constant, ( \frac{d v}{d t} ) is equal to ( r^{2} ).
( \bigcirc ) if ( \frac{d r}{d t} ) is constant, ( \frac{d v}{d t} ) is proportional to ( r^{2} ).
( \bigcirc ) if ( \frac{d r}{d t} ) is constant, ( \frac{d v}{d r} ) is proportional to ( t^{2} ).

Explanation:

Step1: Recall the volume formula of a sphere

The volume formula of a sphere is \(V=\frac{4}{3}\pi r^{3}\).

Step2: Differentiate the volume formula with respect to time \(t\)

Using the chain - rule \(\frac{dV}{dt}=\frac{dV}{dr}\cdot\frac{dr}{dt}\).
Since \(\frac{dV}{dr} = 4\pi r^{2}\) and \(\frac{dr}{dt}=4\).
So \(\frac{dV}{dt}=4\pi r^{2}\times4 = 16\pi r^{2}\).

Step3: Calculate \(\frac{dV}{dt}\) when \(r = 11\)

Substitute \(r = 11\) into \(\frac{dV}{dt}=16\pi r^{2}\).
\(\frac{dV}{dt}=16\pi\times(11)^{2}=16\pi\times121 = 1936\pi\approx1936\times3.14=6080.24\).

Step4: Calculate \(\frac{dV}{dt}\) when \(r = 34\)

Substitute \(r = 34\) into \(\frac{dV}{dt}=16\pi r^{2}\).
\(\frac{dV}{dt}=16\pi\times(34)^{2}=16\pi\times1156 = 18496\pi\approx18496\times3.14 = 57077.44\).

Step5: Analyze part (b)

From \(\frac{dV}{dt}=16\pi r^{2}\) and \(\frac{dr}{dt}\) is constant (\(\frac{dr}{dt} = 4\)).
We can see that \(\frac{dV}{dt}\) is proportional to \(r^{2}\) (because \(\frac{dV}{dt}=16\pi r^{2}\), and \(16\pi\) is a constant).

Answer:

• For \(r = 11\) in: \(6080.24\) \(in^{3}/min\)
• For \(r = 34\) in: \(57077.44\) \(in^{3}/min\)
• For part (b): If \(\frac{dr}{dt}\) is constant, \(\frac{dV}{dt}\) is proportional to \(r^{2}\).