Find the rate at which the water level is rising when the water level is 13 cm.

Question

An inverted pyramid is being filled with water at a constant rate of 75 cubic centimeters per second. The pyramid, at the top, has the shape of a square with sides of length 8 cm, and the height is 15 cm.

Find the rate at which the water level is rising when the water level is 13 cm.

Answer 1

The volume, V, of a square pyramid is:

V = a²h/3 Where h is the height and a is a base edge. The base edge increases in size as the pyramid fills.

The rate of the Volume is given as 75 cm³/s.

The problem is to find the rate, dh/dt, that the water is rising when the water level is 13 cm.

If you look at one side of the pyramid, there are two nested, similar triangles. For the big triangle, h=15 and a = 8. For any height of water level, by similar triangles,

a/h = 8/15 --> a = 8h/15. So our volume formula is now.

V = (8h/15)²h/3 = (64/225) h³/3

dV/dt = (64/225)(3h² dh/dt)/3 = (64/225)h² dh/dt

Solving for dh/dt:

dh/dt = (225/64)(1/h²)dV/dt

when h = 13

dh/dt = (225/(64•13²))(75) = 1.560 cm³/s