Math Problem Statement

A manufacturer of bicycles builds​ racing, touring, and mountain models. The bicycles are made of both steel and aluminum. The company has available 91 comma 200 units of steel and 45 comma 000 units of aluminum. The​ racing, touring, and mountain models need 19​, 24​, and 38 units of​ steel, and 15​, 21​, and 18 units of​ aluminum, respectively. Complete parts​ (a) through​ (d) below. Question content area bottom Part 1 ​(a) How many of each type of bicycle should be made in order to maximize profit if the company makes ​$9 per racing​ bike, ​$13 per touring​ bike, and ​$24 per mountain​ bike? Let x 1 be the number of racing​ bikes, let x 2 be the number touring​ bikes, and let x 3 be the number of mountain bikes. What is the objective​ function? zequals

9x 1plus

13x 2plus

24x 3 ​(Do not include the​ $ symbol in your​ answers.) Part 2 To maximize​ profit, the company should produce

0 racing​ bike(s),

0 touring​ bike(s), and

enter your response here mountain​ bike(s). ​(Simplify your​ answers.) Part 3 ​(b) What is the maximum possible​ profit? The maximum profit is ​$

enter your response here.

Solution

To solve this problem, we need to define and solve a linear programming problem. Here's the breakdown:

Part 1: Objective Function

The objective is to maximize profit, which is given by:

z=9x1+13x2+24x3z = 9x_1 + 13x_2 + 24x_3

where:

  • x1x_1 = number of racing bikes
  • x2x_2 = number of touring bikes
  • x3x_3 = number of mountain bikes
  • $9, $13, and $24 represent the profit from each type of bike, respectively.

Part 2: Constraints

The bikes are made using steel and aluminum, and the company has limitations on the resources:

  • The total available steel is 91,200 units.
  • The total available aluminum is 45,000 units.

Each type of bike requires a specific amount of steel and aluminum:

  • Racing bike: 19 units of steel, 15 units of aluminum.
  • Touring bike: 24 units of steel, 21 units of aluminum.
  • Mountain bike: 38 units of steel, 18 units of aluminum.

Thus, the constraints can be written as:

  1. Steel constraint: 19x1+24x2+38x391,20019x_1 + 24x_2 + 38x_3 \leq 91,200
  2. Aluminum constraint: 15x1+21x2+18x345,00015x_1 + 21x_2 + 18x_3 \leq 45,000
  3. Non-negative constraints: x10,x20,x30x_1 \geq 0, \quad x_2 \geq 0, \quad x_3 \geq 0

Part 3: Solve for Maximum Profit

Using the objective function and the constraints, we solve this using a method like the Simplex Method or any linear programming solver to find the optimal values of x1x_1, x2x_2, and x3x_3 that maximize the profit.

I will now compute the optimal values for x1x_1, x2x_2, and x3x_3.### Part 2: Optimal Production To maximize profit, the company should produce:

  • x1=0x_1 = 0 racing bikes,
  • x2=0x_2 = 0 touring bikes,
  • x3=2400x_3 = 2400 mountain bikes.

Part 3: Maximum Profit

The maximum possible profit is: Maximum profit=57,600\text{Maximum profit} = 57,600

Would you like more details on how this solution was derived, or have any specific questions?

Here are some related questions to expand on this:

  1. How does the availability of resources affect the number of bikes produced?
  2. What happens to the solution if the price of touring bikes increases?
  3. How would the solution change if steel became more limited?
  4. Can you calculate the total amount of steel and aluminum used for this optimal production?
  5. What is the difference in profit if racing bikes were prioritized instead of mountain bikes?

Tip: In linear programming problems, changing the coefficients of the objective function can significantly affect the optimal solution.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Linear Programming
Optimization

Formulas

Objective function: z = 9x1 + 13x2 + 24x3
Steel constraint: 19x1 + 24x2 + 38x3 ≤ 91,200
Aluminum constraint: 15x1 + 21x2 + 18x3 ≤ 45,000
Non-negativity constraints: x1 ≥ 0, x2 ≥ 0, x3 ≥ 0

Theorems

Simplex Method for Linear Programming

Suitable Grade Level

College/Advanced High School