Problem 81 - College Algebra Essentials

Question

The rational expression $$\frac{130 x}{100-x}$$ describes the cost, in millions of dollars, to inoculate \(x\) percent of the population against a particular strain of flu. a. Evaluate the expression for \(x=40, x=80,\) and \(x=90\) Describe the meaning of each evaluation in terms of percentage inoculated and cost. b. For what value of \(x\) is the expression undefined? c. What happens to the cost as \(x\) approaches \(100 \% ?\) How can you interpret this observation?

Step-by-Step Solution

Answer

a) The cost to inoculate 40% of the population is $86.67 million, for 80% it is $520 million, and for 90% is $1170 million. \nb) The expression is undefined when the inoculation percentage is 100%. \nc) As the inoculation percentage approaches 100%, the cost increases dramatically.

Step 1: Evaluate the Expression

Substitute the given percentages (40, 80, and 90) into \(x\) and solve the expression for each value:\n\nFor \(x = 40:\) \nSubstitute \(x = 40\) to obtain the cost: \[ \frac{130 \cdot 40}{100-40} = \frac{5200}{60} = \$86.67 \text{ million}.\]\n\nFor \(x = 80:\) \nSubstitute \(x = 80\) to obtain the cost: \[ \frac{130 \cdot 80}{100-80} = \frac{10400}{20} = \$520 \text{ million}.\]\n\nFor \(x = 90:\) \nSubstitute \(x = 90\) to obtain the cost: \[ \frac{130 \cdot 90}{100-90} = \frac{11700}{10} = \$1170 \text{ million}.\]

Step 2: Determine the Undefined Value

The rational expression is undefined when the denominator equals 0. Therefore, set \(100 - x = 0\), and solve for \(x\). Solution: \(x = 100\). This means that the cost is undefined when 100% of the population is inoculated.

Step 3: Interpret the Cost as the Percentage Approaches 100%

As \(x\) approaches \(100\%\), the denominator of the expression becomes increasingly smaller, making the entire expression (cost) larger. In other words, the cost of inoculation increases dramatically as the percentage of the population inoculated approaches 100%.