SOLUTION: An inverted square pyramid has a height equal to 12 m and a top equal to 6 m. Initially, it contains a water depth of 9m. a. What is the initial volume of the water in the tank?

Algebra ->  Volume -> SOLUTION: An inverted square pyramid has a height equal to 12 m and a top equal to 6 m. Initially, it contains a water depth of 9m. a. What is the initial volume of the water in the tank?      Log On


   

Question 1101394: An inverted square pyramid has a height equal to 12 m and a top equal to 6 m. Initially, it contains a water depth of 9m.
a. What is the initial volume of the water in the tank?

b. If additional water is to be pumped into the tank at the rate of 25 gallons per minute, how many hours will it take to fill the tank?
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


I assume the 6m is the length of a side of the square base....

The volume of the pyramid (in cubic meters) is one-third base times height:
%281%2F3%29%286%5E2%29%2812%29+=+144

When the tank is filled to 3/4 of its depth (9m, out of the total height of 12m), the volume of water is (3/4)^3 of the volume of the whole tank.

That statement uses the powerful general principle concerning similar figures: if the scale factor is a:b, then the ratio of areas is a^2:b^2, and the ratio of volumes is a^3:b^3.

So the volume of water in the tank when the depth is 9m is 27/64 of the total volume: %2827%2F64%29%2A144+=+243%2F4

The volume still remaining in the tank is (1-(27/64)) = 37/64 of the total volume of the tank: %2837%2F64%29%2A144+=+333%2F4

That's the interesting part of the problem.
I leave it to you to convert that volume in cubic meters to gallons, and to find the amount of time it takes to finish filling the tank at 25 gallons per minute.