Question

Water is dripping out of the bottom of a conical container. The cone has a diameter of 8 inches and a height of 8 inches. a) When the depth of the water is 8 inches, what is the radius of the water? b) Write a function for the volume of water left in the container in terms of the radius of the water. c) Write a function for the volume of water left in the container in terms of the height of the water. d) Find the radius of the cone when the volume is 24pi inches^3.

Answer 1

a) if the cone is 8 inches high and it is filled with a depth of 8 inches, then the cone is full of water, and the radius of the water level must be 4 inches.
Is there a typo here?

b)
let the height of the water be h, and the radius of the water level be r
by similar triangles:
h/r = 8/4
h = 2r

Volume of water = (1/3)π r^2 h
= (1/3)π r^2(2r)
= (2/3)π r^3

c) from above , r = h/2
V = (1/3)π r^2 h
= (1/3)π (h^2/4)h
= (1/12)π h^3

d) (1/12)π h^3 = 24π
h^3 = 288
h = appr 6.6 inches