A 1500-liter tank contains 70 liters of a 30% acid solution and you add x liters of a 70% acid solution to the tank. If the function C, which represents the proportion of acid to the total solution in the final mixture, is given by , what percent does the concentration of acid appear to be approaching as x increases without bound? (Ignore the physical constraint that .)
A) 100%
B) 30%
C) 40%
D) 70%
E) 20%

Question

A 1500-liter tank contains 70 liters of a 30% acid solution and you add x liters of a 70% acid solution to the tank. If the function C, which represents the proportion of acid to the total solution in the final mixture, is given by  , what percent does the concentration of acid appear to be approaching as x increases without bound? (Ignore the physical constraint that  .)
A) 100% 
B) 30% 
C) 40% 
D) 70% 
E) 20%

Quizplus · Accepted Answer

First, we can find an expression for the total amount of acid in the final mixture by adding the amount of acid in the initial solution (30% of 70 liters) to the amount of acid added (70% of x liters). This gives a total acid amount of 21 + 0.7x.

Next, we can find an expression for the total amount of solution in the final mixture by adding the initial 1500 liters to the x liters added. This gives a total solution amount of 1500 + x.

Using these expressions, we can plug them into the given function C: 
C(x) = (21 + 0.7x)/(1500 + x)

We want to find the limit of this function as x approaches infinity. We can use L'Hopital's Rule to take the limit of the numerator and denominator separately:

lim as x → ∞ (21 + 0.7x)/(1500 + x)
= lim as x → ∞ 0.7/(1/1500)
= 0.7 * 1500 
= 1050

So the concentration of acid appears to be approaching 1050/1500 = 0.7, or 70%. Therefore, the answer is D.

A 1500-Liter Tank Contains 70 Liters of a 30% Acid