SOLUTION: Officials begin to release water from a full man-made lake at a rate that would empty the lake in 16 weeks, but a river that can fill the lake in 25 weeks is replenishing the lake

Question 1184052: Officials begin to release water from a full man-made lake at a rate that would empty the lake in 16 weeks, but a river that can fill the lake in 25 weeks is replenishing the lake at the same time. How many weeks does it take to empty the lake? Express your answer as a fraction reduced to lowest terms, if needed.
Found 2 solutions by ikleyn, Theo:
Answer by ikleyn(52767)   (Show Source): You can put this solution on YOUR website!
.
Officials begin to release water from a full man-made lake at a rate that would empty the lake in 16 weeks,
but a river that can fill the lake in 25 weeks is replenishing the lake at the same time.
How many weeks does it take to empty the lake? Express your answer as a fraction reduced to lowest terms, if needed.
~~~~~~~~~~~~~

The rate of draining is of the reservoir volume per week. The rate of filling is of the reservoir volume per week. The net rate of draining is the difference - = = of the reservoir volume per week. Hence, the process will be completed in weeks = 44 weeks. ANSWER

Solved.

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Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
rate * time = quantity.

you have 1 lake, so quantity = 1.

formula becomes rate * time = 1

it takes 16 weeks to empty the lake.
rate * 16 = 1
rate = 1/16
this means 1/16 of the lake is empties each week.

it takes 25 weeks to fill up the lake.
rate * 25 = 1
rate = 1/25
this means 1/25 of the lake is replenished each week.

every week, 1/16 week of the lake is removed and 1/25 of the lake is replenished.
the net loss each week is 1/16 - 1/25 = 25/400 - 16/400 = 9/400 of the lake.

if you start with a full lake and have a net loss of 9/400 of the lake each week, the formula of rate * time = quantity becomes 9/400 * time = 1
solve for time to get:
time = 400/9 weeks.

the lake will be empty in 400/9 weeks.

in that time, the amount of lake that is removed is 1/16 * 400/9 = 25/9 times the original amount of the lake.

in that time, the amount of lake that is replenished is 1/25 * 400/9 = 16/9 times the original amount of the lake.

the net amount of lake that is removed in 400/9 weeks is 25/9 - 16/9 = 9/9 = 1 times the mount of water in the lake.

this confirms the solution is correct.

the solution is that it takes 400/9 weeks to empty the lake.

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