Problem 70 - College Algebra

Question

The rational function $$C(x)=\frac{130 x}{100-x}, 0 \leq x<100$$ describes the cost, \(C(x),\) in millions of dollars, to inoculate \(x \%\) of the population against a particular strain of flu. a. Find and interpret \(C(20), C(40), C(60), C(80),\) and \(C(90)\) b. What is the equation of the vertical asymptote? What does this mean in terms of the variables in the function? c. Graph the function.

Step-by-Step Solution

Answer

a. \(C(20) = 32.5, C(40) = 86.67, C(60) = 195, C(80) = 520, C(90) = 1170\). Each output represents the cost in millions of dollars to inoculate the corresponding percentages of the population. b. The vertical asymptote is \(x = 100\), meaning cost rises infinitely as the percentage of population inoculated approaches 100%. c. The graph of the function shows this trend visually.

Step 1: Evaluate the function at given inputs

Plug the values \(x = 20, 40, 60, 80, 90\) into the function \(C(x)\) to find the corresponding costs. For example, to find \(C(20)\), \(C(20) = \frac{130 * 20}{100 - 20} = 32.5\) represents the cost in millions of dollars to inoculate 20% of the population.

Step 2: Determine the vertical asymptote

The denominator of the function cannot be zero, as this would render the function undefined. Thus, for the function \(C(x)=\frac{130x}{100-x}\), 100 - x cannot be zero. This gives \(x = 100\) as the vertical asymptote. This means as the percentage of the population who are inoculated approaches 100%, the cost of inoculating increases indefinitely.

Step 3: Graph the function

The function can be graphed using a graphing tool, taking into account that the cost increases as a greater percentage of the population is inoculated, and that the cost approaches infinity as the percentage inoculated approaches 100%, represented by the vertical asymptote \(x = 100\). The graph starts from the origin, indicating that the cost is zero when no part of the population is inoculated.

Key Concepts

Asymptote Interpretation

To fully grasp the concept of asymptotes in the context of rational functions, one must first understand what an asymptote is. An asymptote is a line that a graph approaches but never actually reaches, signifying that within the context of the function, there's a boundary which the output can approach infinitely but can't cross.

In our example, the rational function is given by C(x) = \(\frac{130x}{100-x}\), and we determine the vertical asymptote by setting the denominator equal to zero and solving for x. Since dividing by zero is undefined in mathematics, the function will not have an output at x = 100, which is thus our vertical asymptote.

This has a real-world interpretation: as the percentage of the population getting inoculated against a particular strain of flu gets closer to 100%, the cost C(x) escalates rapidly, moving toward infinity. This reflects the realistic scenario where the final portions of a population can be increasingly expensive to reach for vaccination due to various factors like accessibility, availability, or resistance.

Graphing Functions

Graphing a function allows us to visualize how the output of that function, or y-value, changes in response to different inputs, or x-values. In algebra, this visualization helps us understand the function’s behavior in a tangible way.

To graph the rational function C(x) = \(\frac{130x}{100-x}\), we plot points for known values of x and connect these points to illustrate the function's progression. For example, finding C(20), C(40), C(60), C(80), and C(90) provides specific costs for inoculating certain percentages of the population and helps us form the graph.

Using a graphing tool, we begin at the origin (0,0), which signifies no cost for no inoculation. The graph steadily rises, reflecting the increasing cost for inoculating more of the population. As we near the vertical asymptote at x = 100, the graph gets steeper and tends toward infinity, symbolic of skyrocketing costs for vaccinating nearly the entire population.

Cost Analysis in Algebra

Cost analysis is a crucial application of algebra in real-world situations. It involves creating functions to model the cost of a service or product given different variables. In our scenario, the function C(x) = \(\frac{130x}{100-x}\) models the cost in millions of dollars to inoculate a percentage of the population.

Conducting a cost analysis with this function means calculating the cost for various values of x, which represents the percentage of the population. For instance, C(20) yields $32.5 million to inoculate 20% of the population. By comparing these costs—C(40), C(60), C(80), and particularly C(90)—we notice how the cost to inoculate increases at a greater rate as we approach complete coverage.

This analysis aids in budget planning and resource allocation for such a large-scale health initiative. Understanding how costs escalate as we strive for higher inoculation rates can inform strategies to achieve the best balance between coverage and cost efficiency.