Gravel is dumping from a conveyor at a rate of 35ft3/min, it forms a pile in the shape of a cone whose base diameter & height are equal. How fast is the height of the pile incr. when it's 15ft high

Answer 1

These questions are all about writing down any relationships we know in a mathematical form

To start they GIVE US the derivative of volume with respect to time.

dV/dt = 35ft³/min

They are asking us for the change in height with respect to time.

dh/dt = ?

We can look up the equation for the volume of a cone

V = πr²h/3

We are told that the diameter is equal to the height, so we can actually get our equation down to a single variable h.

2r = h

r = h/2

Now we have

V = πh³/12

Since we know dV/dt, it makes sense to take the derivative of our equation and set it equal to the value they gave us

dV/dt = 3πh²/12 dh/dt

35 = 3πh²/12 dh/dt

From here, they actually gave us h as well! So let's plug it in

35 = 3π(15)²/12 dh/dt

Simply rearrange and get

0.198 ft/min