Problem 25 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Formulate but do not solve the problem. You will be asked to solve these problems in the next section. The management of Hartman Rent-A-Car has allocated $$\$ 1.5$$ million to buy a fleet of new automobiles consisting of compact, intermediate-size, and full- size cars. Compacts cost $$\ 12,000$$ each, intermediate size cars cost $$\ 18,000$$ each, and full-size cars cost $$\ 24,000$$ each. If Hartman purchases twice as many compacts as intermediate-size cars and the total number of cars to be purchased is 100, determine how many cars of each type will be purchased. (Assume that the entire budget will be used.)

Step-by-Step Solution

Answer

Hartman Rent-A-Car should purchase 60 compact cars, 30 intermediate-size cars, and 10 full-size cars.

Step 1: Define the variables

Let's define three variables to represent the number of cars of each type: \(x\) = number of compact cars \(y\) = number of intermediate-size cars \(z\) = number of full-size cars

Step 2: Write the linear equations

We can write the following linear equations based on the problem conditions: 1. Equation representing the total budget: \(\ 12,000x + 18,000y + 24,000z = 1,500,000\) 2. Equation representing the relation between the number of compact and intermediate-size cars: \(x = 2y\) 3. Equation representing the total number of cars to be purchased: \(x + y + z = 100\)

Step 3: Solve the linear system

Using substitution, we can substitute \(x = 2y\) into equation (3): \(2y + y + z = 100\) Now, we can solve for \(z\): \(3y + z = 100\) Next, we will substitute \(x = 2y\) into equation (1): \(\ 12,000(2y) + 18,000y + 24,000z = 1,500,000\) Now, we have: \(\ 42,000y + 24,000z = 1,500,000\) We can simplify this to: \(\ 7y + 4z = 250\) Now we have 2 equations with 2 variables, \(y\) and \(z\): 1. \(3y + z = 100\) 2. \(7y + 4z = 250\) Now let's solve the linear system using substitution or elimination. We will use elimination: Multiply the first equation by 4 to have same coefficient of z in both equations: \(12y + 4z = 400\) Subtract the first equation from the second equation: \((7y + 4z) - (12y + 4z) = 250 - 400\) This simplifies to: \(-5y = -150\) Now we can solve for y: \(y = 30\) Now we can find z by substituting the value of y in equation 1: \(3(30) + z = 100\) This simplifies to: \(90 + z = 100\) Now we can solve for z: \(z = 10\) Finally, we can find the value of x using the equation \(x = 2y\): \(x = 2(30)\) This gives us the value of x: \(x = 60\)

Step 4: State the answer

Hartman Rent-A-Car should purchase: - 60 compact cars - 30 intermediate-size cars - 10 full-size cars.

Key Concepts

System of Equations

A system of equations is a set of two or more equations that have common variables. Solving these systems involves finding values for the variables that satisfy all the equations simultaneously. In the context of this problem, there are three key equations which need to be solved using the variables defined. To represent the relationship of the costs, the equation \(12,000x + 18,000y + 24,000z = 1,500,000\) was derived, modeling the budget constraint. Another equation, \(x = 2y\), captures the condition that twice as many compact cars will be bought compared to intermediate-size cars. Lastly, the equation \(x + y + z = 100\) represents the total number of cars to be purchased. These equations form a system that describes all conditions of the problem.

Variable Definition

Defining variables is a fundamental step in setting up mathematical models. In our problem, we defined three variables to stand for the types of cars to be purchased. These are:

• \(x =\) the number of compact cars
• \(y =\) the number of intermediate-size cars
• \(z =\) the number of full-size cars

This approach allows us to translate real-world constraints into mathematical language. Properly defining variables ensures precision and clarity in setting up and solving equations. It also helps in ensuring that each equation corresponds accurately to a specific condition described by the problem.

Budget Constraints

Budget constraints are economic limitations that impact decision-making. In this scenario, the total budget allocated for purchasing cars is \$1.5 million. This budget constraint is represented mathematically as \(12,000x + 18,000y + 24,000z = 1,500,000\), where each term reflects the cost of one type of car multiplied by the number of cars purchased. Budget constraints force decision-makers to make strategic choices about resource allocation. By incorporating constraints within equations, we can identify feasible solutions that align with financial limits, ensuring that decisions are practical and abide by the given financial boundaries.