Problem 81 - College Algebra

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

When \(x=40, 80,\) and \(90\), the cost to inoculate x % of the population is approximately 86.67, 520, and 1170 million dollars respectively. The expression is undefined when x equals 100 percent. As x is close to or equal to 100 % the cost to inoculate will rapidly approach infinity suggesting prohibitive costs to inoculate the entirety of the population.

Step 1: Evaluate the expression

Given expression is \(\frac{{130x}}{{100-x}}\), let's substitute the given values (x being 40, 80, and 90 respectively). When \(x=40\), the expression equals \(\frac{{130*40}}{{60}} = 86.67 \), million dollars. Similarly, when \(x=80\), the expression equals \(\frac{{130*80}}{{20}} = 520 \), million dollars, and when \(x=90\), the expression equals \(\frac{{130*90}}{{10}} = 1,170 \), million dollars. Meaning, the cost to inoculate 40%, 80%, and 90% of the population are 86.67, 520, and 1,170 million dollars respectively.

Step 2: Finding undefined value for x

To find the value of x for which the expression is undefined, we need to find the value of x that makes the denominator zero, because division by zero is undefined. So we solve the equation \(100 - x = 0\), which leads to \(x = 100\). The expression is undefined when x equals 100 percent.

Step 3: Interpretation as x approaches 100%

When the value of x is close to 100%, the denominator approaches zero, hence the expression will tent to infinity. This suggests that as the attempt to innoculate the entire population (100%) is approached, the cost will rapidly increase towards infinity. The interpretation is that it would cost nearly an infinite amount of money to inoculate 100% of the population entirely due to increasing marginal costs or logistical difficulties.

Key Concepts

Percentages

A percentage is a way of expressing a number as a fraction of 100. It is often used to describe how a part relates to a whole. In real-life contexts like finance and health, percentages are key in breaking down numbers into understandable segments.
For example, if we say 40% of the class scored an A, it means 40 out of every 100 students did well.

• Percentages help compare relative sizes, especially when dealing with different total amounts.
• They are typically represented by the symbol %.

In the context of the given rational expression, percentages indicate the proportion of the population being inoculated.
Evaluating the expression for specific percentages provides insight into how costs change with different levels of inoculation. As we see, a higher percentage like 90% incurs a much greater cost compared to 40%.

Undefined Expressions

An undefined expression occurs when you attempt to perform an operation that is not mathematically possible. One of the most common causes is division by zero.
In mathematics, any number divided by zero is undefined because it does not result in a finite number or pattern.

• If a rational expression has a denominator that equals zero, the entire expression becomes undefined.
• This is crucial for understanding and solving problems involving rational expressions.

In our exercise, the expression \( \frac{130x}{100-x} \) becomes undefined when \( x \) is 100. That's because substituting 100 makes the denominator zero: \(100 - 100 = 0\).
This means it is mathematically impossible to calculate the cost when attempting to inoculate 100% of the population based on this formula.

Limits in Algebra

Limits help us understand the behavior of a function as it approaches a particular input value, especially where the function might not be defined exactly.
When evaluating limits, we analyze how the function behaves very close to a specific point.

• Limits are fundamental in calculus but are also useful in algebra for understanding trends and behavior in functions.
• They can show us what happens to a function when inputs get very large or very small.

For the rational expression \( \frac{130x}{100-x} \), as \( x \) gets closer to 100, the cost seems to shoot upwards towards infinity.
This is described as the limit approaching infinity as \( x \to 100 \). In practical terms, it indicates that trying to inoculate 100% of the population would be extremely costly, theoretically requiring infinite resources, reflecting the steep logistical and marginal cost increases.