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.
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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.

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:

\[
z = 9x_1 + 13x_2 + 24x_3
\]

where:
- $ x_1 $ = number of racing bikes
- $ x_2 $ = number of touring bikes
- $ x_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:
   \[
   19x_1 + 24x_2 + 38x_3 \leq 91,200
   \]
2. Aluminum constraint:
   \[
   15x_1 + 21x_2 + 18x_3 \leq 45,000
   \]
3. Non-negative constraints:
   \[
   x_1 \geq 0, \quad x_2 \geq 0, \quad x_3 \geq 0
   \]

### 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 $x_1$, $x_2$, and $x_3$ that maximize the profit.

I will now compute the optimal values for $x_1$, $x_2$, and $x_3$.### Part 2: Optimal Production
To maximize profit, the company should produce:
- $ x_1 = 0 $ racing bikes,
- $ x_2 = 0 $ touring bikes,
- $ x_3 = 2400 $ mountain bikes.

### Part 3: Maximum Profit
The maximum possible profit is:
\[
\text{Maximum profit} = 57,600
\]

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.

A manufacturer of bicycles builds racing, touring, and mountain models. The bicycles are made of both steel and aluminum. The company has available 91,200 units of steel and 45,000 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. (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? (b) What is the maximum possible profit?

Solve a linear programming problem to maximize profit for a bicycle manufacturer. Learn how to allocate 91,200 units of steel and 45,000 units of aluminum to racing, touring, and mountain bikes for optimal production. Includes the use of the Simplex Method for linear optimization.

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