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

Given:

Inverted square pyramid filled at constant rate of 75 cm^3/s

Sides of square base (at top) are 8 cm.

Height is 15 cm.

Find: Rate the water level is rising when water level is 13 cm.

Solution:

Let s = side length and r = s/2

As water level rises, both the height and side increase.

The height and side are proportional to the structure.

15/4 = h/r

h = (15/4)*r

h = (15/4)*(s/2)

h = (15/8)*s Relates height and side.

s = (8/15)*h

V = (1/3)*h*s^2 Vol. of square pyramid

V = (1/3)*h*[ (8/15)*h ]*2 Vol. in terms of h only

V = (1/3)*(64/225)*h^3

V = (64/675)*h^3

V' = (64/675)*(3)*(h^2)*(h') Take derivative of both sides

75 = (64/675)*(3)*(13^2)*(h') Use given information

Solve for h'

h' = (75)*(675/64) / [ (3)*(13^2) ]

h' = (50625/64) / (507)

h' = 16875 / 10816

h' = 1.56019 cm/s