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Officials begin to release water from a full man-made lake at a rate that would empty the lake in 14
14
weeks, but a river that can fill the lake in 25
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

Sagot :

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Answer:

[tex]\frac{350}{11} weeks[/tex]

Step-by-step explanation:

Let V be the volume of water in the lake and w be the number of weeks to empty the lake

Since it takes 14 weeks to empty the lake by water release, the empty rate = V/14

But at the same time the river is replenishing the lake at a rate of V/25 since it will take 25 weeks to fill the empty lake

  • Net empty rate =  Empty rate - replenishment rate =  [tex]\frac{V}{14} - \frac{V}{25} = \frac{V}{w}[/tex]
  • Divide the equation by V to eliminate V giving

               [tex]\frac{1}{14} - \frac{1}{25} = \frac{1}{w}[/tex]

  • Multiply by 350 (14 x 25) to eliminate the denominators

              [tex]25 - 14 = \frac{350}{w}[/tex]  ==> [tex]11 = \frac{350}{w}[/tex]

  • Multiply both sides by w to cancel the denominator w

                  11w = 350

  • Divide both sides by 11 to get

                    w = [tex]\frac{350}{11 }[/tex] weeks

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