Question

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

Answer 1

It takes 20 weeks to empty the lake

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)=(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)=(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:

`(0.05)/(1) = (1)/(total\ of\ weeks)  \\total\ of\ weeks = (1)/(0.05)=(1)/((1)/(20) )\\  total\ of\ weeks = 20`

Thus, It takes 20 weeks to empty the lake.