Write an example where rational function is used in real-life situation
The correct answer and explanation is:
A real-life example of a rational function can be found in business, specifically in pricing and cost analysis. Consider a company that manufactures a product, and it wants to determine the cost per unit when producing a certain number of units. The total cost function might involve both fixed costs and variable costs.
For example, assume the fixed costs for a factory are \$100,000, and the variable cost per unit produced is \$10. The cost to produce $x$ units can be expressed as:
$$
C(x) = \frac{100,000}{x} + 10
$$
Here, the rational function represents the relationship between the cost per unit and the number of units produced. The term $\frac{100,000}{x}$ reflects the decreasing cost per unit as production increases, because the fixed costs are spread over more units. The $10$ is the variable cost per unit, which remains constant regardless of the production level.
Explanation:
This rational function is used in business to make decisions regarding optimal production levels. The total cost function shows how costs behave as production increases. In particular, the term $\frac{100,000}{x}$ demonstrates the principle of economies of scale, where fixed costs decrease per unit as production increases. If the company produces just a few units, the cost per unit will be very high, as the fixed costs are distributed over fewer units. But as production increases, each unit “shares” a smaller portion of the fixed costs, lowering the overall per-unit cost.
In practice, rational functions like this one are crucial in determining pricing strategies, understanding profit margins, and identifying break-even points, where total revenue equals total costs. It also helps businesses determine at what point producing more units will no longer result in significant cost savings, guiding decision-makers on efficient production levels.