Answer to Question #336374 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.

Sand is being dropped at a rate of 10 cubic feet per minute onto a conical pile. If

the height of the pile is always twice the base radius, at what rate is the height

increasing when the pile is 8 feet high?

PLEASE ANSWER MY QUESTION QUICKLY!!

DEADLINE : 05/03/2022 11 : 00 PM

Answer 1

$$V=\dfrac{1}{3}\pi r^2h$$

Given $h=2r$

$$V=\dfrac{\pi}{12}h^3$$

Differentiate both sides with respect to $t$

$$\dfrac{dV}{dt}=\dfrac{\pi}{4}h^2(\dfrac{dh}{dt})$$

Then

$$\dfrac{dh}{dt}=\dfrac{4}{\pi h^2}(\dfrac{dV}{dt})$$

When the pile is 8 feet high

$$\dfrac{dh}{dt}=\dfrac{4}{\pi (8ft)^2}(10{ft}^3/min)$$

$$=\dfrac{5}{8\pi}ft/min\approx0.199ft/min$$

The height is increasing at rate of 0.199 feet per minute when the pile is 8 feet high.