the radius of a spherical balloon is increasing at a rate of 4 centimeters per minute. how fast is the…
the radius of a spherical balloon is increasing at a rate of 4 centimeters per minute. how fast is the volume changing, in cubic centimeters per minute, when the radius is 8 centimeters? note: the volume of a sphere is given by v=(4/3)πr³. rate of change of volume, in cubic centimeters per minute, =
Answer
Explanation:
Step1: Differentiate volume formula
We have $V=\frac{4}{3}\pi r^{3}$. Using the power - rule for differentiation $\frac{d}{dr}(r^{n})=nr^{n - 1}$, we get $\frac{dV}{dr}=4\pi r^{2}$.
Step2: Apply chain - rule
By the chain - rule, $\frac{dV}{dt}=\frac{dV}{dr}\cdot\frac{dr}{dt}$. We know that $\frac{dr}{dt}=4$ cm/min.
Step3: Substitute values
Substitute $r = 8$ cm and $\frac{dr}{dt}=4$ cm/min into $\frac{dV}{dt}=4\pi r^{2}\cdot\frac{dr}{dt}$. So $\frac{dV}{dt}=4\pi(8)^{2}\times4$.
Step4: Calculate result
$\frac{dV}{dt}=4\pi\times64\times4=1024\pi$ $cm^{3}/min$.
Answer:
$1024\pi$