A company manufactures two models of snowboards, standard and deluxe. Each deluxe model requires 28 hours to produce, and 60 units of material. Each standard model requires 14 hours to produce and 50 units of material. The company has 1050 production hours available and 2750 units of material in stock. The deluxe model sells for $252 and the standard model sells for $180.
What amount of each model should be produced and sold to obtain the maximum revenue?
deluxe models
standard models
What is the maximum revenue? $

This problem involves finding the number of deluxe and standard snowboards to produce in order to maximize the company's revenue. It can be modeled as a linear programming problem, where we want to maximize revenue subject to certain constraints.

### Variables
Let:
- $ x $ represent the number of **deluxe** models produced.
- $ y $ represent the number of **standard** models produced.

### Objective Function (Maximizing Revenue)
The revenue from selling $ x $ deluxe models and $ y $ standard models is:
\[
\text{Revenue} = 252x + 180y
\]
We want to maximize this.

### Constraints
1. **Production Hours:** The company has 1050 production hours available. Each deluxe model requires 28 hours, and each standard model requires 14 hours. This gives us the constraint:
\[
28x + 14y \leq 1050
\]

2. **Material Units:** The company has 2750 units of material available. Each deluxe model requires 60 units, and each standard model requires 50 units. This gives us the constraint:
\[
60x + 50y \leq 2750
\]

3. **Non-negativity:** The number of snowboards produced cannot be negative, so:
\[
x \geq 0, \quad y \geq 0
\]

### Problem Summary
Maximize $ 252x + 180y $ subject to:
\[
28x + 14y \leq 1050
\]
\[
60x + 50y \leq 2750
\]
\[
x \geq 0, \quad y \geq 0
\]

Now I'll solve this using linear programming.To maximize revenue, the company should produce and sell:
- **25 deluxe models**
- **25 standard models**

The maximum revenue they can achieve is **$10,800**.

Would you like further details or clarifications?

Here are 5 related questions to expand your understanding:
1. How would changes in production hours affect the number of snowboards produced?
2. What happens if the price of the deluxe model increases? How does that affect the solution?
3. What if the material supply is reduced to 2500 units? How would that impact production?
4. Can the company optimize for maximum profit instead of revenue by including production costs?
5. How would the solution change if the standard model sold for $200 instead of $180?

**Tip:** In linear programming, constraints define the feasible region, while the objective function determines the optimal point within that region.

A company manufactures two models of snowboards, standard and deluxe. Each deluxe model requires 28 hours to produce, and 60 units of material. Each standard model requires 14 hours to produce and 50 units of material. The company has 1050 production hours available and 2750 units of material in stock. The deluxe model sells for $252 and the standard model sells for $180. What amount of each model should be produced and sold to obtain the maximum revenue?

Learn how to maximize revenue for a company producing two models of snowboards using linear programming. This problem involves constraints on production hours and material, and optimizes revenue for deluxe and standard models. Suitable for Grades 11-12.

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