Question 277017: Because of an anticipated heavy rainstorm, the water level in a reservoir must be lowered by 1 foot. Opening Spillway A lowers the level by this amount in 4 hours, whereas opening the smaller spillway B does the job in 6 hours. How long will it take to lower the water level by 1 foot if both spillways are opening?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! rate * time = units
the number of units is equal to 1 (lowering the level in the reservoir by 1 foot)
Spillway A can do the job in 4 hours.
The rate at which spillway A can do the job is equal to 1/4 which means that spillway A can lower 1/4 of a foot per hour.
Spillway B does the job in 6 hours.
The rate at which spillway B can do the job is equal to 1/6 which means that spillway A can lower 1/6 of a foot per hour.
If you open both spillways, then their rates are additive.
This means that:
(1/4 + 1/6) * T = 1 where T is the number of hours it will take when both work together.
Remove parentheses to get:
(1/4)*T + (1/6)*T = 1
Multiply both sides of this equation by 24 to get:
6*T + 4*T = 24
Simplify to get:
10*T = 24
Divide both sides of this equation by 10 to get:
T = 2.4
It should take 2.4 hours for both working together.
Plug into original equation to see if this is accurate.
(1/4)*2.4 + (1/6)*2.4 = 1.
Simplify to get:
.6 + .4 = 1 which is true confirming the values are good.
In the following graph, y represents the rate, and x represents the time.
You can see that
when x = 4, y = 1/4 which is the rate of spillway A.
when x = 6, y = 1/6 which is the rate of spillway B.
when x = 2.4, y = 10/24 which is the combined rate of spillway A and B.
1/4 + 1/6 = 10/24.
The horizontal lines on the graph are 10/24, 1/4, and 1/6 respectively going from top to bottom.
They correspond to the x values of 2.4, 4, and 6 respectively going from left to right.
When spillway A and spillway B work together, they can lower the level by 1 foot in 2.4 hours.
Since Rate * Time = Work (expressed in units produced), then:
R * 2.4 = 1 which results in R = 1/2.5 which is the same as 10/24 which is the combined rate of spillway A and spillway B.
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