Question

A chemist has 500 mL of a 30% acid solution. She adds x milliliters of a 10% acid solution. Which statement is true about the graph of the function that represents the concentration of the final solution?
The horizontal asymptote y = 30 means that the final concentration will always be less than 30%.
The horizontal asymptote y = 10 means that the final concentration will always be greater than 10%.
The vertical asymptote x = 500 means that the volume of the solution will always be greater than 500 mL.
The vertical asymptote x = 0 means that the volume of the solution will always be greater than 0 mL.

Answer 1

Let f(x) be the final concentration (as a decimal). The amount of acid is 150 + 0.1x mL and the total volume is 500 + x mL, so
f(x) = (150 + 0.1x)/(500 + x).

As x → ∞, f(x) → 0.1 (10%), so the horizontal asymptote is y = 10. Also
f(x) − 0.1 = 100/(500 + x) > 0 for x ≥ 0,
so f(x) is always greater than 10% for any nonnegative x. Therefore the second statement is true; the others are false.