This is a APGP question

Question

A chemist has a vat containing 50 litres of chemical. A sample of 10 litres is drawn from the vat and replaced with 10 litres of water. After complete mixing, another sample of 10 litres of the mixture is drawn and replaced with 10 litres of water. The process is repeated many times. Let 𝑐𝑛 denote the amount of chemical in the vat just before the 𝑛th sample is drawn and let 𝑑𝑛 denote the amount of chemical contained in the 𝑛th sample.

(i) Write down an equation connecting (a) 𝑐𝑛 and 𝑐𝑛+1 (b) 𝑐𝑛 and 𝑑𝑛.

(ii) Find the amount of chemical contained in the vat just before the 9th sample is drawn?

(iii) Calculate the total amount of chemical removed from the vat after the 9th sample is drawn.

Answer 1

Let C_o = the original amount of chemical (50 litres)

C₁ = the amount of chemical after after 10 litres is removed and 10 liters of water is added the first time

and so on.

The recursive formula appears to be C_n+1 = C_n/1.25

Starting with the original amount of 50, after removing 10 litres and adding 10 litres water, the concentration is 40.

or C₁ = C_o/1.25 = 50/1.25 = 40

Then next time, (C₂ - C₁)/C₂ = -.25 or 1.25C₂ = C₁ or C₂ = C₁/1.25 = C_o/(1.25)²

Now you can see that C_n = Co/(1.25)ⁿ Then C₉ = C_o/(1.25)⁹ = 50/(1.25)⁹ = 6.71 litres

This should help you complete the three questions!