Problem 81 - Algebra and Trigonometry

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 of inoculation for \(x=40\) is 86.67 million dollars, for \(x=80\) is 520 million dollars and for \(x=90\) is 1300 million dollars. b. The expression is undefined at \(x=100\). c. As \(x\) approaches \(100 \% \), the cost of inoculation increases indefinitely. This implies that is becomes extremely expensive to inoculate the entire population.

Step 1: Evaluating the expression

The expression given is \( \frac{130 x}{100-x} \). Start by substituting the values \(x=40, x=80,\) and \(x=90\) into the expression and calculate the results. These results represent the cost, in millions of dollars, to inoculate the respective percentage of the population.

Step 2: Finding undefined point

The expression is undefined if the denominator \(100-x\) equals zero (since it is not defined to divide by zero). So to find the undefined point, you need to solve equation \(100-x=0\).

Step 3: Analyzing the function's behavior

See what happens with the expression as \(x\) approaches \(100 \% \). This requires knowledge of limit analysis. As \(x\) approaches \(100\), the denominator of the rational expression will approach \(0\), which would make the whole expression go to infinity.

Step 4: Interpreting the results

Finally, interpret the results in context. The cost to inoculate the population against the flu virus increases as the percentage of the population inoculated increases. As \(x\) approaches \(100 \% \), the cost increases indefinitely, which denotes that it becomes extremely expensive to inoculate the entire population.