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.

Explanation:

Step1: Find the function for the concentration of the final solution

The amount of acid in the 30% solution is $0.3\times500 = 150$ mL. The amount of acid in the 10% solution is $0.1x$ mL. The total volume of the final solution is $(500 + x)$ mL. So the concentration $C(x)$ of the final solution is $C(x)=\frac{150 + 0.1x}{500+x}$.

Step2: Analyze horizontal asymptotes

As $x\to\infty$, we use the rule for rational - functions. For $y=\frac{150 + 0.1x}{500+x}$, divide both the numerator and denominator by $x$: $y=\frac{\frac{150}{x}+0.1}{\frac{500}{x}+1}$. As $x\to\infty$, $\frac{150}{x}\to0$ and $\frac{500}{x}\to0$. So $y\to0.1$. The horizontal asymptote $y = 10$ (since $0.1$ is 10% in percentage form) means that as we add more and more of the 10% solution, the final concentration will approach 10% but will always be greater than 10% because we start with a 30% solution.

Step3: Analyze vertical asymptotes

The function $C(x)=\frac{150 + 0.1x}{500+x}$ is undefined when the denominator is zero. Setting $500 + x=0$, we get $x=- 500$. But in the context of adding a volume of solution, $x\geq0$. The vertical - asymptote analysis in the given options is incorrect.

Answer:

The horizontal asymptote $y = 10$ means that the final concentration will always be greater than 10%.