The volume of the balloon is [tex]2352\pi cc/min[/tex]
Explanation:
The radius of the balloon is increasing at a rate of 3 cm/min.
To determine the volume of the balloon when the radius is 14 cm, we shall use the formula [tex]V=\frac{4}{3} \pi r^3[/tex]
The rate of change of r with respect to time t is given by,
[tex]\frac{d}{d t}(r)=3 \mathrm{cm} / \mathrm{minute}[/tex]
Now, we shall determine the [tex]\frac{d}{d t}(V)[/tex]
[tex]\begin{aligned}\frac{d}{d t}(V) &=\frac{d}{d t}\left(\frac{4}{3} \pi r^{3}\right) \\&=\frac{4}{3} \pi\left(3 r^{2}\right)\frac{d}{d t}(r) \\&=4 \pi r^{2}\frac{d}{d t}(r)\end{aligned}[/tex]
Now, we shall determine the [tex]\frac{d}{d t}(V)[/tex] at [tex]r=14 \mathrm{cm}[/tex] and substituting [tex]\frac{d}{d t}(r)=3 \mathrm{cm} / \mathrm{minute}[/tex], we get,
[tex]\begin{aligned}\left(\frac{d V}{d t}\right)_{r=14} &=4 \pi r^{2} \frac{d}{d t}(r)\\&=4 \pi(14)^{2} (3)\\&=4 \pi 196 (3)\\&=2352\end{aligned}[/tex]
Thus, The volume of the balloon is [tex]2352\pi cc/min[/tex]
