the radius of a spherical balloon is increasing at the rate of 0.7 cm/minute. how fast is the volume…

the radius of a spherical balloon is increasing at the rate of 0.7 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.)

the radius of a spherical balloon is increasing at the rate of 0.7 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.)

Answer

Explanation:

Step1: Recall volume formula for sphere

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

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 know that $\frac{dr}{dt}=0.7$ cm/min and $r = 7.4$ cm. Substitute these values into the derivative formula: $\frac{dV}{dt}=4\pi(7.4)^{2}(0.7)$.

Step4: Calculate the result

First, calculate $(7.4)^{2}=54.76$. Then $4\pi(54.76)(0.7)=4\times3.14\times54.76\times0.7$. $4\times3.14\times54.76\times0.7 = 12.56\times54.76\times0.7=688.8856\times0.7 = 482.21992\approx482.2$ $cm^{3}/min$.

Answer:

$482.2$