a spherical balloon is inflated and its volume increases at a rate of 24 in.³/min. what is the rate of…

Explanation:

Step1: Recall volume formula for sphere

The volume formula of a sphere is $V = \frac{4}{3}\pi r^{3}$, where $V$ is the volume and $r$ is the radius.

Step2: Differentiate with respect to time $t$

Using the chain - rule, $\frac{dV}{dt}=4\pi r^{2}\frac{dr}{dt}$.

Step3: Solve for $\frac{dr}{dt}$

We know that $\frac{dV}{dt} = 24$ in³/min and $r = 10$ in. Rearranging the equation $\frac{dV}{dt}=4\pi r^{2}\frac{dr}{dt}$ for $\frac{dr}{dt}$, we get $\frac{dr}{dt}=\frac{\frac{dV}{dt}}{4\pi r^{2}}$.

Step4: Substitute the given values

Substitute $\frac{dV}{dt}=24$ and $r = 10$ into the formula: $\frac{dr}{dt}=\frac{24}{4\pi\times(10)^{2}}=\frac{24}{400\pi}=\frac{3}{50\pi}$.

Step5: Calculate the value

$\frac{3}{50\pi}\approx\frac{3}{50\times3.14159}\approx0.019$ in/min.

Answer:

$0.019$ in/min