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

Answer 1

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

Step-by-step 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=(4)/(3) \pi r^3`

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

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

Now, we shall determine the
`(d)/(d t)(V)`

`\begin{aligned}(d)/(d t)(V) &=(d)/(d t)\left((4)/(3) \pi r^(3)\right) \\&=(4)/(3) \pi\left(3 r^(2)\right)(d)/(d t)(r) \\&=4 \pi r^(2)(d)/(d t)(r)\end{aligned}`

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

`\begin{aligned}\left((d V)/(d t)\right)_(r=14) &=4 \pi r^(2) (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`