How fast is the water level rising?

Question

a paper cup has the shape of a cone with height 8cm and radius 3 cm at the top. Water is poured into the cup at a rate of 2 cm^3/s. How fast is the water level rising when the water level is 6 cm deep?

Answer 1

Let h = depth of water at time t

      r = radius of surface of water at time t

     V = volume of water at time t

 

Given:  dV/dt = 2

Find:  dh/dt when h = 6

 

The volume of the cone is (1/3)πr²h

By similar triangles, 8/3 = h/r, so 8r = 3h

Therefore, r = (3/8)h

 

So, V = (1/3)π(3/8h)²h = (3/64)πh³

 

Differentiate with respect to t:  

dV/dt = (3/64)π3h²(dh/dt)

dV/dt = (9/64)πh²(dh/dt)

 

2 = (9/64)π(6²)(dh/dt)

2 = (81/16)π(dh/dt)

 

dh/dt = (32)/(81π) ≈ 0.126 cm/sec