Question

cles and compares find that the cost of pollution control increases along with the percentage of pollutants to be removed in a situation. suppose that the cost c, in dollars, of removing p% of the pollutants from a chemical spill is given below. complete parts (a) through (d).

c(p)=\frac{32000}{100 - p}

(c) sketch a graph of c. choose the correct graph below.

(d) can the company or city afford to remove 100% of the pollutants due to this spill? explain.

a. yes, as the percent of pollutants cleaned up, p, approaches 100, the cost, c, arrives at $32,000 which is affordable.

b. no, as the percent of pollutants cleaned up, p, approaches 100, the cost, c, approaches infinity which is not affordable.

Explanation:

Step1: Analyze the function \(C(p)=\frac{32000}{100 - p}\)

The denominator \(100 - p\) cannot be zero. When \(p = 100\), the function is undefined. As \(p\) approaches \(100\) from the left (\(p\to100^{-}\)), we consider the limit \(\lim_{p\to100^{-}}\frac{32000}{100 - p}\). Let \(x=100 - p\), when \(p\to100^{-}\), \(x\to0^{+}\). Then \(\lim_{x\to0^{+}}\frac{32000}{x}=\infty\).

Step2: Evaluate the options

Option A is incorrect because \(\lim_{p\to100^{-}}C(p)
eq32000\). Option B is correct since as \(p\) approaches \(100\) (the percentage of pollutants cleaned up), the cost \(C(p)=\frac{32000}{100 - p}\) approaches infinity (because the denominator approaches \(0\) from the positive side) and an infinite - cost is not affordable.

Answer:

B. No, as the percent of pollutants cleaned up, \(p\), approaches \(100\), the cost, \(C\), approaches infinity which is not affordable.