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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.

Ask by Turnbull Bowers. in the United States
Mar 17,2025

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Answer

**A. Horizontal Asymptote** There are no horizontal asymptotes. **B. Intercepts** - \( x \)-intercept: \( x=0 \) - \( y \)-intercept: \( y=0 \)

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Bonus Knowledge

As \( x \) approaches 1 in the function \( P(x) = \frac{11x}{1-x} \), the denominator approaches 0, causing \( P(x) \) to grow without bound. Therefore, the horizontal asymptote occurs at \( y = 11 \) as \( x \) approaches 0, meaning that when the contamination is minimal, the monetary contribution stabilizes. For the \( x \)-intercept, we find where \( P(x) = 0 \). Setting \( \frac{11x}{1-x} = 0 \), we see that \( 11x = 0 \) leads to \( x = 0 \). Thus, the \( x \)-intercept is at \( x=0 \), indicating that when no contaminants are removed, the required contribution is zero!

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