SOLUTION: An inverted square pyramid has a height equal to 8 meters and a top edge to 3 meters. Initially, it contains water to the depth of 5 meters. a. What is the initial volume of the

Algebra ->  Volume -> SOLUTION: An inverted square pyramid has a height equal to 8 meters and a top edge to 3 meters. Initially, it contains water to the depth of 5 meters. a. What is the initial volume of the       Log On


   

Question 575067: An inverted square pyramid has a height equal to 8 meters and a top edge to 3 meters. Initially, it contains water to the depth of 5 meters.
a. What is the initial volume of the water in the tank?
b. If the additional water is to be pumped into the tank at the rate of 20 gallons per minute, how many hours will it take to fill the tank?
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
For a square base pyramid
s=length of base side (in meters)
h=height (in meters
V=volume (in cubic meters
V=%281%2F3%29s%5E2%2Ah
The inverted pyramid tank has A volume of
V=%281%2F3%293%5E2%2A8=%281%2F3%29%2A9%2A8=24 cubic meters
a. The initial volume of water in the tank is the volume of a similar pyramid with
a height of 5 meters. Since the this pyramid and the 8 meter high pyramid are similar, with a height ratio of
5%2F8, the ratio of their volumes is %285%2F8%29%5E3
So the initial volume of water in the tank is
%285%2F8%29%5E3%2A24=75%2F64=5.859375 cubic meters
b.Since the tank's volume was 24 cubic meters, the amount of water that must be added to fill the tank (in cubic meters) is
24-75%2F64=24-5.859375=18.140625
One cubic meter is 1000 liters and one US gallon is approximately 3.79L, so we approximate and convert
%2818.14m%5E3%29%281000L%2Fm%5E3%29%281gallon%2F3.79L%29=approximately 4786 gallons
At 20 gallons per minute, with 60 minutes per hour, the hours needed to fill the tank are
%284786gallons%29%281minute%2F20gallons%29%281hour%2F60minutes%29= approximately 3.99hours, so we'll say that thew answer is 4 hours.