Formulate a linear programming problem that can be used to solve the following question.
A plane delivers cargo in two types of crates between two destinations. The light crate is 16 cubic feet in volume and 1100 pounds in weight, and earns $11 in revenue. Each heavy crate is 20 cubic feet in volume and 700 pounds in weight, and earns $25 in revenue. The plane has available at most 1296 cubic feet and 78300 pounds for the crates. Finally, at least twice the number of light crates as the heavy ones must be shipped. Find the number of crates of each type of cargo to ship in order to maximize revenue.
x = 
---Select---

y = 
---Select---



---Select---
F = 
 	(objective function)
Subject to	
 	(volume)
 	(weight)
 	(ratio)
x 
---Select---
 0, y 
---Select---
 0	(nonnegativity constraint)

To formulate this problem as a linear programming (LP) problem, let's define variables and constraints based on the details given. Here's the setup:

### Variables:
- Let $ x $ be the number of light crates to be shipped.
- Let $ y $ be the number of heavy crates to be shipped.

### Objective Function:
We aim to maximize revenue, which is calculated based on the revenue per crate:
\[
F = 11x + 25y
\]
Where:
- $ 11x $ represents the revenue from light crates.
- $ 25y $ represents the revenue from heavy crates.

### Constraints:
1. **Volume Constraint**:
   The total volume occupied by the crates cannot exceed the available volume of 1296 cubic feet:
   \[
   16x + 20y \leq 1296
   \]
2. **Weight Constraint**:
   The total weight of the crates cannot exceed the available weight limit of 78,300 pounds:
   \[
   1100x + 700y \leq 78300
   \]
3. **Ratio Constraint**:
   There must be at least twice as many light crates as heavy crates:
   \[
   x \geq 2y
   \]
4. **Non-negativity Constraint**:
   The number of crates cannot be negative:
   \[
   x \geq 0, \quad y \geq 0
   \]

### Complete Linear Programming Problem:

Maximize:
\[
F = 11x + 25y
\]
Subject to:
\[
16x + 20y \leq 1296 \quad \text{(volume constraint)}
\]
\[
1100x + 700y \leq 78300 \quad \text{(weight constraint)}
\]
\[
x \geq 2y \quad \text{(ratio constraint)}
\]
\[
x \geq 0, \quad y \geq 0 \quad \text{(non-negativity)}
\]

This formulation provides the mathematical structure needed to solve the problem using linear programming techniques.

Would you like more details or have any questions?

Here are five related questions:
1. How do you solve a linear programming problem using the graphical method?
2. What are slack variables, and how do they work in linear programming?
3. How would the problem change if there were different revenue amounts for each crate?
4. What if the plane had additional restrictions, such as a maximum number of crates allowed?
5. How would you model this problem if more types of crates were introduced?

**Tip:** In linear programming, always check if the feasible region is bounded to ensure there is an optimal solution.

Formulate a linear programming problem that can be used to solve the following question. A plane delivers cargo in two types of crates between two destinations. The light crate is 16 cubic feet in volume and 1100 pounds in weight, and earns $11 in revenue. Each heavy crate is 20 cubic feet in volume and 700 pounds in weight, and earns $25 in revenue. The plane has available at most 1296 cubic feet and 78300 pounds for the crates. Finally, at least twice the number of light crates as the heavy ones must be shipped. Find the number of crates of each type of cargo to ship in order to maximize revenue.

Learn how to formulate and solve a linear programming problem involving the optimization of cargo delivery. This problem involves maximizing revenue by shipping light and heavy crates with specific volume and weight constraints. Perfect for students studying optimization, inequalities, and linear programming at an advanced high school or college level.

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