In Silicon Valley, a number of computer firms were found to be contaminating underground water supplies with toxic chemicals stored in leaking underground containers. A water quality control agency ordered the companies to take immediate corrective action and contribute to a monetary pool for the testing and cleanup of the underground contamination. Suppose that the required monetary pool (in millions of dollars) is given by the following function, where $$ x $$ is the percentage (expressed as a decimal fraction) of the total contaminant removed. Complete parts (A) through (E). $$ P(x)=\frac{11 x}{1-x} \quad 0 \leq x

Question

In Silicon Valley, a number of computer firms were found to be contaminating underground water supplies with toxic chemicals stored in leaking underground containers. A water quality control agency ordered the companies to take immediate corrective action and contribute to a monetary pool for the testing and cleanup of the underground contamination. Suppose that the required monetary pool (in millions of dollars) is given by the following function, where $$ x $$ is the percentage (expressed as a decimal fraction) of the total contaminant removed. Complete parts (A) through (E). $$ P(x)=\frac{11 x}{1-x} \quad 0 \leq x<1 $$ Find the horizontal asymptote(s), if any. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The horizontal asymptote(s) is (are) $$ y= $$ (Simplify your answer. Use a comma to separate answers as needed) B. There are no horizontal asymptotes. (D) Find the $$ x $$ - and $$ y $$-intercepts. Find the $$ x $$-intercept(s). Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The $$ x $$-intercept(s) is (are) $$ x= $$ (Simplify your answer. Use a comma to separate answers as needed.) B. There are no $$ x $$-intercepts. B.

UpStudy AI · Accepted Answer

**Step 1. Find the Horizontal Asymptote**

The function is  
$$
P(x)=\frac{11x}{1-x}.
$$

Since the numerator and the denominator are both first–degree polynomials, the horizontal asymptote is given by the ratio of the leading coefficients. However, note that the denominator is $$1-x$$. We can rewrite the function as  
$$
P(x)=\frac{11x}{1-x}=\frac{11x}{-(x-1)}=-\frac{11x}{x-1}.
$$

Now, the leading coefficient of the numerator (in the rewritten form) is $$-11$$ and the leading coefficient of the denominator is $$1$$. Thus, the horizontal asymptote is  
$$
y=\frac{-11}{1}=-11.
$$

**Step 2. Find the $$x$$-intercepts**

The $$x$$-intercepts occur when $$P(x)=0$$. Set the function equal to zero:
$$
\frac{11x}{1-x}=0.
$$

A rational function equals zero when the numerator equals zero (provided the denominator is not zero). Set the numerator equal to zero:
$$
11x=0 \quad\Longrightarrow\quad x=0.
$$

Checking the denominator at $$x=0$$:
$$
1-x = 1-0=1 \neq 0,
$$
so $$x=0$$ is valid.

Thus, the $$x$$-intercept is  
$$
x=0.
$$

**Answers:**

A. The horizontal asymptote is $$y=-11$$.

D. The $$x$$-intercept is $$x=0$$.
A. The horizontal asymptote is $$ y = -11 $$. D. The $$ x $$-intercept is $$ x = 0 $$.

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