An inverted pyramid is being filled

Question

An inverted pyramid is being filled with water at a constant rate of 65 cubic centimeters per second. The pyramid, at the top, has the shape of a square with sides of length 7 cm, and the height is 10 cm.

Find the rate at which the water level is rising when the water level is 3 cm.

Answer 1

The volume of the inverted pyramid is (1/3)x²y where x(t) is the side of the square and y(t) is the height of the pyramid at the water level. (Hence, it is the volume of water added by time t.

We want to express x as x(y) so that V can be V(y). The ratio of x to y will always be 7/10 because of similarity.

V = (1/3)(7y/10)²y = (49/300)y³

dV/dt = (49/100)y²(dy/dt) and dy/dt = (100/49)(dV/dt)/(3 cm)² = 14.74 cm/s with dV/dt=65 cm³/s

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