related rates

Question

a paper cup as the shape of a cone with height 10 cm and radium 3 cm at the top. if water is poured into the cup at a rate of 2cm^3/s, how fast is the water level rising when water is 5 cm deep

Answer 1

Let V = volume of water in the cup at time t

 

     r = radius of water's surface at time t

 

     h = depth of water at time t

 

Given:  dV/dt = 2  

 

Find:  dh/dt when h = 5

 

Solution:  V = (1/3)πr²h

 

              Using similar triangles, 3/10 = r/h

 

              So, 10r = 3h.   r = (3/10)h

 

So, V = (1/3)π[(3/10)h]²h

 

     V = (3/100)h³

 

   dV/dt = (9/100)h²(dh/dt)

 

        2 = (9/100)(5²)(dh/dt)

 

        2 = (9/4)(dh/dt)

 

   dh/dt = 8/9  cm/sec