the radius of a sheridan balloon is increasing at a rate of 3 centimeters per minute. how fast is the volume changing when the radius is 14

Question

the radius of a sheridan balloon is increasing at a rate of 3 centimeters per minute. how fast is the volume changing when the radius is 14 centimeters? v=(4/3)(pi)r^3

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Anna 3 weeks 2021-12-26T08:13:21+00:00 1 Answer 0 views 0

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    2021-12-26T08:14:59+00:00

    The volume of the balloon is 2352\pi cc/min

    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 V=\frac{4}{3} \pi r^3

    The rate of change of r with respect to time t is given by,

    \frac{d}{d t}(r)=3 \mathrm{cm} / \mathrm{minute}

    Now, we shall determine the \frac{d}{d t}(V)

    \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}

    Now, we shall determine the \frac{d}{d t}(V) at r=14 \mathrm{cm} and substituting \frac{d}{d t}(r)=3 \mathrm{cm} / \mathrm{minute}, we get,

    \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}

    Thus, The volume of the balloon is 2352\pi cc/min

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