Question

Sand is pouring out of a pipe and is forming a conical pile on the ground. The radius of the pile is increasing at a rate of 2 feet per day. The height is always 1 2 of the radius. When the pile has a radius of 18 feet, a) the Volume is 3053.628059 Correct cubic feet b) the rate of change of the VOLUME is Incorrect cubic feet per day..

Answer 1

Answer:

$a)\ V=3053.628059 \ ft^3$

$b)\ V'=1017.876\ ft^3/day$

Explanation:

Rate of Change

The volume of a cone of radius r and height h is given by

$\displaystyle V=\frac{\pi r^2h}{3}$

The height is said to be 1/2 of the radius, thus

$\displaystyle V=\frac{\pi r^2\cdot r}{2\cdot 3}$

$\displaystyle V=\frac{\pi r^3}{6}$

a) Knowing r=18 feet, the volume is

$\displaystyle V=\frac{\pi 18^3}{6}$

$V=3053.628059 \ ft^3$

b) The rate of change of the volume is computed by taking the derivative of both sides respect to the time

$\displaystyle V'=3\frac{\pi r^2}{6}r'$

$\displaystyle V'=\frac{\pi r^2}{2}r'$

Where r' is the given rate of change of the radius: 2 feet/day.

Now we compute

$\displaystyle V'=\frac{\pi 18^2}{2}\cdot 2=1017.876$

$\boxed{V'=1017.876\ ft^3/day}$