Question 1084499: the diameter of a hose is inversely proportional to the time it takes to fill a tank.
if the diameter of hose is 1 inch, and the filling rate is 10 ft per second, how much water will be in the tank after 1.75 seconds?
appreciate your help,
Gul
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the diameter of the hose is inversely proportional to the time it takes to fill the tank.
this means that when the diameter of the hose is larger, it takes less time to fill the tank.
the formula for indirect variation is y = k/x
k is the constant of variation.
it always stays the same once it is set up initially based on the problem statement.
you are given that it takes 1.75 seconds to fill the tank when the diameter of the hose is 1 inch.
the formula of y = k/x becomes 1.75 = k/1
solve for k to get k = 1.75
that's the constant of variation.
when the diameter of the hose is 2 inches, it should take half the time to fill the tank, and it does:
y = k/x becomes y = 1.75/2 = .875 seconds.
the diameter of the hose is doubled and it takes half the time to fill the tank.
how does this relate to the rate at which the hose is delivering water?
the problem states that the 1 inch hose delivers 10 cubic feet of water per second.
rate * time = quantity
the 1 inch hose delivers water at the rate of 10 cubic feet per second for 1.75 seconds, so the formula becomes:
10 cubic feet per second * 1.75 seconds = 17.5 cubic feet of water.
if the tank is full, then the capacity of the tank has to be 17.5 cubic feet of water.
the 1 inch hose delivers 17.5 cubic feet of water in 1.75 seconds.
note that i used cubic feet instead of feet because we're talking volume of water and volume is measured in cubic units.
now, our indirect ratio says that the time it takes is indirectly proportional to the diameter of the hose, and we figured out that the constant of variation was 1.75.
when the diameter of the hose was 1 inch, it took 1.75 seconds to fill the tank.
when the diameter of the hose was 2 inches, the formula said that y = 1.75 / 2 and we got that it would then take .875 seconds to fill the tank.
the rate * time = quantity formula that was 10 * 1.75 = 17.5 now becomes:
x * .875 = 17.5
x is the rate.
.875 is the time
17.5 is the cubic feet of water that is the capacity of the tank.
solve for x to get x = 17.5 / .875 = 20
when the diameter of the hose is 2 inches, it is delivering 20 cubic feet of water in one second.
the diameter of the hose was doubled and the time it took to fill the tank became half and the rate that the hose delivered the water was doubled.
makes sense to me.
hopefully it will make sense to you.
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