Rates of Change

Question

Milk is poured into a bowl at a constant rate of 15.0 cm³/s. The bowl has a circular cross section, but does not have a uniform diameter. (That is, different horizontal cross sections taken at different heights of the bowl have different diameters.)

(a) As the milk fills the bowl, the milk level reaches a point where the diameter of the bowl is d₁ = 1.25 cm.What is the rate (in cm/s) at which the milk level rises at this diameter?

Answer 1

Let v = 15cm³/s be the rate at which volume of milk increases in the container,

h(t) be the height of the milk at time t,

r(h) be the radius of the container at height h.

Consider a particular time t₀, h₀ = h(t₀), r₀ = r(h₀).

The area of the milk surface at time t₀ is 2πr₀².

In a very small time period dt,

the amount of milk added will be vdt,

and that will raise the height by an amount dh, so that the volume

2πr₀²dh = vdt

From that

dh/dt = v/2πr₀²

We are to calculate dh/dt at a point in time t₀ when r₀ = r(t₀) = 1.25/2 cm.

Substituting:

dh/dt = 15 /(2π(1.25/2)²) = 6.1 cm/s