# Limits and Derivatives

This is a homework question where I don't quite understand what I am being asked to do:

A tank contains $5000$ L of pure water. Brine that contains $30$ g of salt per litre of water is pumped into the tank at a rate of $25$ L/min. Show that the concentration of salt after t minutes (in grams per litre) is:

$$C(t)= 30t/200+t$$

I am not sure what I am being asked to do?

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-1. Not a math question, just an inquiry about the purpose of a math question. –  user26649 May 21 '12 at 21:55
@Farhad: It is not about the purpose of the question; it’s asking what the question means. –  Brian M. Scott May 21 '12 at 21:57
Of course it is a math question. A school math question and not a scaled down research question. But M.SE is for all levels of math. –  Michael Greinecker May 21 '12 at 21:59
Is $C(t)= 30t/(20+t)$, instead of $C(t)= 30t/200+t$? –  Américo Tavares May 21 '12 at 22:06
@FarhadYusufali That downvote is not called for. It is worrying that you think this isn't a math question. –  Pedro Tamaroff May 21 '12 at 22:19

$30$ grams of salt per litre and $25$ litres per minute mean that salt is being pumped into the tank at a rate of $30\cdot25=750$ grams per minute. After $t$ minutes this will add $750t$ grams of salt to the tank. Since the water started out pure, the total amount of salt in the tank after $t$ minutes must be $750t$ grams.

The concentration in grams per litre is the total number of grams of salt in the tank divided by the number of litres of solution. How many liters of solution will be in the tank after $t$ minutes? Express that in terms of $t$, and you have everything you need.

(By the way, the answer that you list is incorrect; it contains a typo.)

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Your expression seems to be incorrect. The correct expression seems to be $$C(t) = \frac{30t}{20 + t}$$.

Now, let us say 1 minute has passed after the pumps were started. So, you added 25L of water to the tank. That 25L of water contains 30g per litre. Thereby, in total it contains $30 \times 25 = 750$g of salt. The total amount of water in the tank after 1 minute is 525L. Therefore, the concentration of salt (amount of salt per litre) in the tank after 1 minute is given by $\frac{750}{525} = \frac{30}{21}$. This matches the above expression.

Now, you should be able to generalize this.

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I made a typo in the question, the initial volume of water is actually 5000L. The chapter leading up to this question was on Limits, and when I came to this question I, for some reason, thought it would be something to do with Limits. Anyways, thanks for the response. –  Kurt May 21 '12 at 22:16

salt after $t$ minutes = $30 \cdot 25 \cdot t$

water after $t$ minutes = $500 + 25 t$

compute concentration $30 \cdot 25 \cdot t \over 500 + 25 t$

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