Answer to Question #336375 in Calculus for John

Question

Related Rates: Problem Solving

Direction: Solve each word problem involving a related rates. Make a sketch of your work

if necessary, and then apply differentiation.

A spherical snow ball is made so that its volume is increasing at a rate of 8 cubic

feet per minute. Find the rate at which the radius is increasing when the snowball

is 4 feet in diameter.

PLEASE ANSWER MY QUESTION QUICKLY!!

DEADLINE : 05/03/2022 11 : 00 PM

Answer 1

$$V=\dfrac{4}{3}\pi R^3$$

Differentiate both sides with respect to $t$

$$\dfrac{dV}{dt}=\dfrac{4\pi}{3}(3R^2)(\dfrac{dR}{dt})$$

Then

$$\dfrac{dR}{dt}=\dfrac{1}{4\pi R^2}(\dfrac{dV}{dt})$$

Given $\dfrac{dV}{dt}=8{ft}^3/min, D=4ft.$

$$\dfrac{dR}{dt}=\dfrac{1}{4\pi (4ft/2)^2}(8{ft}^3/min)=\dfrac{1}{2\pi}ft/min$$ $$\approx0.159 ft/min$$

The radius is increasing at rate of 0.159 feet per minute when the diameter is 4 feet.