Question

the radius of a spherical balloon is increasing at the rate of 0.9 cm/minute. how fast is the volume changing when the radius is 7.4 cm? the volume is changing at a rate of cm³/minute. (type an integer or a decimal. round to one decimal place as needed.)

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: Substitute given values

We are given that $\frac{dr}{dt}=0.9$ cm/min and $r = 7.4$ cm. Substitute these values into the derivative formula: $\frac{dV}{dt}=4\pi(7.4)^{2}(0.9)$.

Step4: Calculate the result

First, calculate $(7.4)^{2}=54.76$. Then, $4\pi\times54.76\times0.9=4\pi\times49.284$. Since $\pi\approx3.14$, we have $4\times3.14\times49.284 = 619.0$.

Answer:

$619.0$