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Question
A tank contains 5000 litres of pure water. Brine (very salty water) that contains 30 grams of salt per litre of water is pumped into the tank at a rate of 25 litres per minute. The concentration of salt water after t minutes (in grams per litre) is C(t) = `(30"t")/(200 + "t")`. What happens to the concentration as t → ∞?
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Solution
Given the concentration of saltwater after t minutes is C(t) = `(30"t")/(200 + "t")`
To find the concentration of saltwater after t → ∞
`lim_("t" -> oo) "C"("t") = lim_("t" -> oo) (30"t")/(200 + "t")`
= `lim_("t" -> oo) (30"t")/("t"(200/"t" + 1))`
= `lim_("t" -> oo) (30"t")/(200/"t" + 1)`
= `30/(0 + 1)`
`lim_("t" -> oo) "C"("t")` = 30
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