[Calc] Sequences | Suppose you start with one liter of vinegar and repeatedly remove 0.06 L, replace with water, mix, and repeat.

Question

Suppose you start with one liter of vinegar and repeatedly remove 0.06 L, replace with water, mix, and repeat.

a. Find a formula for the concentration after  steps.

       

b. After how many steps does the mixture contain less than 12% vinegar?

     

Answer 1

a₁ = 1

a₂ = 0.94 a₁

a₃ = (0.94) a₂ = (0.94)² a₁

a_n = (0.94)^n-1

0.12 > (0.94)^n-1

Solution left for you to do.

Answer 2

If C₀ = 1 then we may write:

C₁=(1-0.06) C₀

C₂=(1-0.06) C₁ = (1-0.06)² C₀

.....

C_n = (1-0.06)ⁿ C₀

Therefore if (1-0.06)ⁿ C₀ < 12/100, we can take logarithm of both sides and obtain

n log(94/100) < log(12/100), which implies that n > 34.