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melamori03 [73]
3 years ago
14

Officials begin to release water from a full man-made lake at a rate that would empty the lake in 10 weeks, but a river that can

fill the lake in 20 weeks is replenishing the lake at the same time. How many weeks does it take to empty the lake
Mathematics
1 answer:
Romashka [77]3 years ago
5 0

Answer:

It takes 20 weeks to empty the lake

Step-by-step explanation:

Some rate, let's call it R_{empty} empties the lake in 10 weeks and a river (with rate R_{river} can fill the lake in 20 weeks. The idea behind the solution of this type of problem is always set the rate to the same units. Let's see this:

R_{empty}=\frac{1\ lake}{10\ weeks} which implies that this rate empties 0.1 of the lake per week, thus, R_{empty}=0.1/week.

Using the same idea, we can obtain that the filling rate of the river is:

R_{river}=\frac{1\ lake}{20\ weeks} = 0.05 /week

Since both rates are acting at the same time, the total rate of flowing water (either in or out the lake) would be given by:

R_{total}=R_{empty}+R_{river}\\R_{total}= -0.1 + 0.05=-0.05/week, which means that the lake is being empty at a rate of -0.05 week and using the rule of three:

\frac{0.05}{1} = \frac{1}{total\ of\ weeks}  \\total\ of\ weeks = \frac{1}{0.05}=\frac{1}{\frac{1}{20} }\\  total\ of\ weeks = 20

Thus, It takes 20 weeks to empty the lake.

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