Answer: he volume of a pyramid is 1/3 Bh, where B is the area of the base and h is the height. Since the base is square, B = s2, where s is the side of the base. So the volume is 1/3 s2h.
We know that the full pyramid has a height of 11 cm and a side of the base is 8 cm. Also, no matter what the height of the water is, the surface of the water will form a square which has the same ratio to the depth of the water as a side of the top of the pyramid does to the full height of the pyramid. So that means that no matter what the depth of the water, s = 8/11 h.
So that means that the formula for the volume of the pyramid formed by the water is V = 1/3 (8/11 h)2 h = 64/363 h3.
Differentiate this as a function of time to get the rate of change of the volume of water:
V = 64/363 h3
dV/dt = 3 (64/363) h2 dh/dt = 192/363 h2 dh/dt
But we know that the volume is changing at a rate of 65 cc / sec. So at any point in time,
65 = 192/363 h2 dh/dt
65 * 363 / 192 = h2 dh/dt
23595 / 192 = h2 dh/dt
(23595 / 192) / h2 = dh/dt
When the water level (h) is 8, that means:
(23595 / 192) / 64 = dh/dt
1.92 = dh/dt
So the water level is rising at the rate of 1.92 cm per second when the water level is 8 cm.
Explanation: