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Chapter 5: Problem 15

A courier company makes deliveries with two different trucks. Truck A costs \(\$ 0.62 / \mathrm{mi}\) to operate and truck \(\mathrm{B}\) costs \(\$ 0.50 / \mathrm{mi}\) to operate. Write an objective function \(z=f(x, y)\) that represents the total cost for driving truck A for \(x\) miles and driving truck \(\mathrm{B}\) for \(y\) miles.

Short Answer

Expert verified
z = 0.62x + 0.50y

Step by step solution

01

- Identify Cost per Mile for Each Truck

Truck A costs \(\$ 0.62 \) per mile to operate, and Truck B costs \(\$ 0.50 \) per mile to operate. These values will be used in the objective function.
02

- Define Variables for Miles Driven

Let \( x \) represent the number of miles driven by Truck A, and let \( y \) represent the number of miles driven by Truck B.
03

- Form the Objective Function

The total cost can be represented as \( z \), which is the sum of the costs for both trucks. Therefore, the objective function will be \( z = 0.62x + 0.50y \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

cost function
A cost function helps us measure the total cost of performing certain activities, such as operating trucks for deliveries. In this problem, we have two trucks with different costs per mile. Truck A costs \(\$ 0.62\) per mile, whereas Truck B costs \(\$ 0.50\) per mile. The goal is to create a formula that will calculate the total cost based on the miles driven by both trucks. By doing so, we can manage and predict expenses more accurately.
To form the cost function, let's use the given costs and represent them mathematically. We know the cost for each mile for both trucks. Combining these costs with the respective miles driven, we can express the total cost as a formula, known as the objective function.
variables in algebra
Variables are symbols that represent unknown values or quantities in algebra. They enable us to create general formulas and equations. In this exercise, two variables are used to represent the miles driven by each truck:
  • \( x \): represents the number of miles driven by Truck A
  • \( y \): represents the number of miles driven by Truck B

By using variables, we can create a general formula for the total cost, which will be valid for any number of miles driven by the trucks. These variables are placeholders for the actual values, allowing the formula to be flexible and applicable in different scenarios.
linear equations
A linear equation is a mathematical expression that models a relationship between two variables, where each term is linear, meaning it has no exponents other than 1.
The objective function given in the solution is: \[ z = 0.62x + 0.50y \]
This equation is linear because both \( x \) and \( y \) are to the first power, and the coefficients (0.62 and 0.50) are constants. The equation represents a straight line when graphed on a coordinate plane. This linear form simplifies solving and understanding the relationship between the total cost and the miles driven by each truck. By using this linear equation, we can easily substitute different values of \( x \) and \( y \) to calculate the total cost.

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Most popular questions from this chapter

A couple has \(\$ 60,000\) to invest for retirement. They plan to put \(x\) dollars in stocks and \(y\) dollars in bonds. For parts (a)-(d), write an inequality to represent the given statement. a. The total amount invested is at most \(\$ 60,000\). b. The couple considers stocks a riskier investment, so they want to invest at least twice as much in bonds as in stocks. c. The amount invested in stocks cannot be negative. d. The amount invested in bonds cannot be negative. e. Graph the solution set to the system of inequalities from parts (a)-(d).

Use a graphing utility to approximate the solution(s) to the system of equations. Round the coordinates to 3 decimal places. $$ \begin{array}{l} x^{2}+y^{2}=32 \\ y=0.8 x^{2}-9.2 \end{array} $$

Josh makes \(\$ 24 /\) hr tutoring chemistry and \(\$ 20 / \mathrm{hr}\) tutoring math. Let \(x\) represent the number of hours per week he spends tutoring chemistry. Let \(y\) represent the number of hours per week he spends tutoring math. a. Write an objective function representing his weekly income for tutoring \(x\) hours of chemistry and \(y\) hours of math. b. The time that Josh devotes to tutoring is limited by the following constraints. Write a system of inequalities representing the constraints. \- The number of hours spent tutoring each subject cannot be negative. \- Due to the academic demands of his own classes he tutors at most \(18 \mathrm{hr}\) per week. \- The tutoring center requires that he tutors math at least 4 hr per week. \- The demand for math tutors is greater than the demand for chemistry tutors. Therefore, the number of hours he spends tutoring math must be at least twice the number of hours he spends tutoring chemistry. c. Graph the system of inequalities represented by the constraints. d. Find the vertices of the feasible region. e. Test the objective function at each vertex. f. How many hours tutoring math and how many hours tutoring chemistry should Josh work to maximize his income? g. What is the maximum income? h. Explain why Josh's maximum income is found at a point on the line \(x+y=18\).

A furniture manufacturer builds tables. The cost for materials and labor to build a kitchen table is \(\$ 240\) and the profit is \(\$ 160 .\) The cost to build a dining room table is \(\$ 320\) and the profit is \(\$ 240\). (See Examples \(2-3)\) Let \(x\) represent the number of kitchen tables produced per month. Let \(y\) represent the number of dining room tables produced per month. a. Write an objective function representing the monthly profit for producing and selling \(x\) kitchen tables and \(y\) dining room tables. b. The manufacturing process is subject to the following constraints. Write a system of inequalities representing the constraints. \- The number of each type of table cannot be negative. \- Due to labor and equipment restrictions, the company can build at most 120 kitchen tables. \- The company can build at most 90 dining room tables. \- The company does not want to exceed a monthly cost of \(\$ 48,000\). c. Graph the system of inequalities represented by the constraints. d. Find the vertices of the feasible region. e. Test the objective function at each vertex. f. How many kitchen tables and how many dining room tables should be produced to maximize profit? (Assume that all tables produced will be sold.) g. What is the maximum profit?

Solve the system using any method. $$ \begin{array}{l} 2 x-7 y=2400 \\ -4 x+1800=y \end{array} $$
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