Find the rate at which the water level is rising when the water level is 4 cm.

Question

An inverted pyramid is being filled with water at a constant rate of 55 cubic centimeters per second. The pyramid, at the top, has the shape of a square with sides of length 3 cm, and the height is 9 cm. (The formula for volume of a pyramid is V=lwℎ/3).

Find the rate at which the water level is rising when the water level is 4 cm. Round your answer to the nearest tenth. (Hint: You may need similar triangles.)

________  cm/sec

Answer 1

Let's first simplify that formula for volume by relating height and the side length. So since the sides at the top are 3 cm and the height is 9 cm, we can think of the triangles with general height h to have a side length of h/3 because they create a similar triangle to the filled triangle. And since side length is for length and width, the formula becomes V = (h/3)(h/3)h/3 = h³/27. That's what we'll use in this related rates problem.

Now that 55 cm³/sec is the change in volume with respect to time or ^dV/_dt. We are looking for the rate the height is rising which could be written as ^dh/_dt. And we know that the height of the water is currently 4 cm above the bottom. So we'll take the derivative of the volume equation above and plug in all the values we know.

^dV/_dt = h²/27 ^dh/_dt

55 = (4²/9)(^dh/_dt)

^dh/_dt = = 55*9/16 = 30.9375 cm/sec.