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QuestionA lake empties in 16 weeks and fills in 25 weeks. How long to empty it with both processes? Give your answer as a fraction.

Studdy Solution

STEP 1

Assumptions1. The lake is initially full. . The water is being released from the lake at a rate that would empty the lake in16 weeks.
3. A river is replenishing the lake at a rate that can fill the lake in25 weeks.
4. The rates of water release and replenishment are constant.

STEP 2

First, we need to understand the rates at which the water is being released and replenished. The rate of water release is1 lake per16 weeks, and the rate of replenishment is1 lake per25 weeks.

STEP 3

We can express these rates as fractions of the lake per week.
Rate of release=116 lake/weekRate\, of\, release = \frac{1}{16} \,lake/weekRate of replenishment=125 lake/weekRate\, of\, replenishment = \frac{1}{25} \,lake/week

STEP 4

The net rate of water release is the rate of release minus the rate of replenishment.
Net rate of release=Rate of release−Rate of replenishmentNet\, rate\, of\, release = Rate\, of\, release - Rate\, of\, replenishment

STEP 5

Plug in the values for the rates of release and replenishment to calculate the net rate of release.
Net rate of release=116−125Net\, rate\, of\, release = \frac{1}{16} - \frac{1}{25}

STEP 6

To subtract these fractions, we need to find a common denominator. The least common multiple of16 and25 is400.

STEP 7

Convert the fractions to have the common denominator.
116=25400\frac{1}{16} = \frac{25}{400}125=16400\frac{1}{25} = \frac{16}{400}

STEP 8

Subtract the fractions to find the net rate of release.
Net rate of release=25400−16400=400 lake/weekNet\, rate\, of\, release = \frac{25}{400} - \frac{16}{400} = \frac{}{400} \,lake/week

STEP 9

Now that we have the net rate of release, we can find the time it takes to empty the lake. This is the reciprocal of the net rate of release.
Time to empty the lake=Net rate of releaseTime\, to\, empty\, the\, lake = \frac{}{Net\, rate\, of\, release}

STEP 10

Plug in the value for the net rate of release to calculate the time to empty the lake.
Time to empty the lake=9400Time\, to\, empty\, the\, lake = \frac{}{\frac{9}{400}}

STEP 11

Calculate the time to empty the lake.
Time to empty the lake=9400=4009 weeksTime\, to\, empty\, the\, lake = \frac{}{\frac{9}{400}} = \frac{400}{9} \,weeksIt takes 4009\frac{400}{9} weeks to empty the lake.

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