Question

RELATED RATES QUESTION!
Gravel is being dumped from a conveyor belt at a rate of 30 cubic feet per minute. It forms a pile in the shape of a right circular cone whose base diameter and height are always the same. How fast is the height of the pile increasing when the pile is 12 feet high?

Answer 1

Cone...what equation relates the volume, radius, and height?

Answer 2

\[1/3*\pi*r^2*h\]

Answer 3

You are a listening student. Now, differentiate it implicitly.

Answer 4

can i replace the r with h/2 since the problem says that D and h are always the same?

Answer 5

Try that.

Answer 6

Yes! you are thinking, replace r with h/2 so we don't have two unknowns to solve for! I didn't even see that at first.

Answer 7

i get 2.62 but its wrong. ):

Answer 8

Wait a minute let me solve.

Answer 9

ok(:

Answer 10

I am getting dh/dt = 0.2653 feet / min. when h = 12 feet.

Answer 11

thats right! how did you get that? :O

Answer 12

V = 1/3pir^2h
r = h/2
V=1/3pi(h^2)/4 * h
V=pi/12h^3
dV/dt = pi/12 * 3h^2 * dh/dt
30 = pi/4h^2 * dh/dt
h = 12
solve for dh/dt

Answer 13

\[V=\frac{ 1 }{ 3 }\left( \frac{ h }{ 2 } \right)^2h \rightarrow V=\frac{ \pi h^3 }{ 12 }\rightarrow \frac{ dV }{ dt }=\frac{ 3\pi h^2 }{ 12 }\frac{ dh }{ dt }\]
Did you derive it correctly?