Problem 55 - Algebra for College Students

Question

Solve each problem. Average cost of an SUV. Mercedes-Benz spent \(\$ 700\) million to design its new SUV (Motor Trend, www.motortrend.com). If it costs \(\$ 25,000\) to manufacture each SUV, then the average cost per vehicle in dollars when \(x\) vehicles are manufactured is given by the rational function $$A(x)=\frac{25,000 x+700,000,000}{x}$$ a) What is the horizontal asymptote for the graph of this function? b) What is the average cost per vehicle when \(50,000\) vehicles are made? c) For what number of vehicles is the average cost \(\$ 30,000 ?\) d) Graph this function for \(x\) ranging from 0 to \(100,000\).

Step-by-Step Solution

Answer

The horizontal asymptote is \(y = 25,000\). At 50,000 vehicles, the average cost is \$39,000. For an average cost of \$30,000, 140,000 vehicles must be made.

Step 1 - Find the Horizontal Asymptote

To determine the horizontal asymptote of the function, examine the degrees of the numerator and the denominator. Since both have degree 1, the horizontal asymptote is the ratio of the leading coefficients. That is, \ \(A(x) = \frac{25,000x + 700,000,000}{x} \), the asymptote is \(y = \frac{25000}{1} = 25000\). So the horizontal asymptote is \(y = 25000\).

Step 2 - Calculate Average Cost for 50,000 Vehicles

Substitute \(x = 50,000\) into the average cost formula \ \(A(x) = \frac{25,000x + 700,000,000}{x} \). \ \(A(50,000) = \frac{25,000(50,000) + 700,000,000}{50,000} \). Simplifying, we get \ \(A(50,000) = \frac{1,250,000,000 + 700,000,000}{50,000} = \frac{1,950,000,000}{50,000} = 39,000\). Thus, the average cost per vehicle when 50,000 vehicles are manufactured is \$39,000\.

Step 3 - Solve for Number of Vehicles to Achieve Average Cost of $30,000

Set the average cost formula equal to 30,000 and solve for \(x\): \ \(30,000 = \frac{25,000x + 700,000,000}{x} \). Multiply both sides by \(x\) to clear the denominator: \ \(30,000x = 25,000x + 700,000,000 \, 30,000x - 25,000x = 700,000,000 \, 5,000x = 700,000,000 \, x = \frac{700,000,000}{5,000} = 140,000\). Thus, 140,000 vehicles must be manufactured for the average cost to be \$30,000\.

Step 4 - Graph the Function

To plot the function \ \(A(x) = \frac{25,000x + 700,000,000}{x} \) over the range \(0 \leq x \leq 100,000\), note that as \(x\) increases, the average cost approaches the horizontal asymptote \(y = 25,000\). The graph initially decreases sharply and then levels off near the asymptote as \(x\) increases.

Key Concepts

Rational Functions

In mathematics, a rational function is a ratio of two polynomials. In our exercise, the function representing the average cost, \[A(x) = \frac{25,000x + 700,000,000}{x}\], is a great example of a rational function. The numerator and denominator are polynomial expressions. Rational functions can represent real-world situations where relationships are proportional and often involve rates or averages. Understanding these functions is crucial for analyzing cost behaviors, probabilities, and many other applications in different fields.

Horizontal Asymptote

A horizontal asymptote is a horizontal line that the graph of the function approaches as \(x\) tends towards infinity. For the given average cost function \[A(x) = \frac{25,000x + 700,000,000}{x},\] the degrees of both numerator and denominator are equal. Hence, the horizontal asymptote is found by dividing the leading coefficients. Here, it is \[y = \frac{25,000}{1} = 25,000.\] This means that as more vehicles are manufactured (as \(x\) grows very large), the average cost per vehicle tends to approach \$25,000\ – but it will never reach or exceed this value exactly.

Cost Analysis

Cost analysis involves understanding the different components and behaviors of costs to make better economic decisions. In the given exercise, the cost analysis revolves around calculating the average cost per vehicle when certain numbers of SUVs are manufactured. For instance, making 50,000 vehicles results in an average cost per vehicle of \$39,000.\ To get this average, substitute \(x = 50,000\) into \[A(x) = \frac{25,000(50,000) + 700,000,000}{50,000} \] and simplify. Through this careful analysis, we can determine how the average cost changes with different production levels and infer optimal production numbers.

Graphing Functions

Graphing is a powerful way to visualize the behavior of functions. For the average cost function \[A(x) = \frac{25,000x + 700,000,000}{x},\] graphing helps in understanding how the cost changes as production scales up. When graphing, note key features like the horizontal asymptote at \$25,000,\ which indicates the long-term trend of average cost. Initially, the graph drops sharply as fixed design costs are allocated over more units, then it levels out as production continues to increase. By graphing, you can clearly see how the cost per unit declines and thus make better decisions related to manufacturing quantities.