a spherical balloon is inflated at a rate of 21 cm³/min. at what rate is the diameter of the balloon…

Explanation:

Step1: Recall volume formula for sphere

The volume formula for a sphere in terms of diameter $d$ is $V=\frac{4}{3}\pi(\frac{d}{2})^3=\frac{1}{6}\pi d^{3}$.

Step2: Differentiate with respect to time $t$

Using the chain - rule, $\frac{dV}{dt}=\frac{1}{6}\pi\times3d^{2}\frac{dd}{dt}=\frac{\pi}{2}d^{2}\frac{dd}{dt}$.

Step3: Substitute given values

We know that $\frac{dV}{dt} = 21$ cm³/min and $d = 3$ cm. Substitute these values into the equation $\frac{dV}{dt}=\frac{\pi}{2}d^{2}\frac{dd}{dt}$: $21=\frac{\pi}{2}(3)^{2}\frac{dd}{dt}$. $21=\frac{9\pi}{2}\frac{dd}{dt}$.

Step4: Solve for $\frac{dd}{dt}$

$\frac{dd}{dt}=\frac{21\times2}{9\pi}=\frac{14}{3\pi}$ cm/min.

Answer:

$\frac{14}{3\pi}$ cm/min