Salt is poured from a container at 10 cm³ s-¹ and it formed a conical pile whose height at any time is 1/5 the radius of the above. At what rate is the height of the cone increasing when the height is 2 cm form the base of the cone?

Explanation:

Volume of a cone:

We have and we want to find when the height is 2 cm.

We can see in our equation for the volume of a cone that we have three variables: V, r, and h.

Since we only have dV/dt and dh/dt, we can rewrite the equation in terms of h only.

We are given that the height of the cone is 1/5 the radius at any given time, 1/5r, so we can write this as r = 5h.

Plug this value for r into the volume formula:

Differentiate this equation with respect to time t.

Plug known values into the equation and solve for dh/dt.

Divide both sides by 100π to solve for dh/dt.

The height of the cone is increasing at a rate of 1/10π cm per second.