https://file.aiquickdraw.com/mathbot/file/1728986949768lqsw5c0x.jpg

This problem can be solved using linear programming. Let's define the variables and constraints to maximize the profit.

### Step 1: Define Variables
Let:
- $ x $ be the number of economy blend packages produced.
- $ y $ be the number of superior blend packages produced.

### Step 2: Define the Coffee Usage in Each Blend
Each economy blend package uses:
- 2 ounces of A grade coffee.
- 10 ounces of B grade coffee.

Each superior blend package uses:
- 9 ounces of A grade coffee.
- 2 ounces of B grade coffee.

### Step 3: Constraints (Coffee Availability)
- The total amount of A grade coffee available is 1,728 ounces.
- The total amount of B grade coffee available is 1,760 ounces.

We can express these as:
1. **A grade coffee constraint:**
   \[
   2x + 9y \leq 1728
   \]
   (Total ounces of A grade coffee used by economy and superior blend packages must be less than or equal to 1,728 ounces.)

2. **B grade coffee constraint:**
   \[
   10x + 2y \leq 1760
   \]
   (Total ounces of B grade coffee used by economy and superior blend packages must be less than or equal to 1,760 ounces.)

### Step 4: Objective Function (Profit Maximization)
The profit per economy blend is \$3, and the profit per superior blend is \$4. The objective is to maximize the total profit:
\[
\text{Maximize } 3x + 4y
\]

### Step 5: Non-negativity Constraints
Both $ x $ and $ y $ must be non-negative:
\[
x \geq 0, \quad y \geq 0
\]

### Step 6: Solve the System
We need to solve the system of inequalities:
\[
\begin{aligned}
2x + 9y & \leq 1728 \\
10x + 2y & \leq 1760 \\
x & \geq 0 \\
y & \geq 0
\end{aligned}
\]
to find the values of $ x $ and $ y $ that maximize the profit function $ 3x + 4y $.

I'll calculate the solution for you next.The optimal solution is to produce:
- $ x = 144 $ economy blend packages.
- $ y = 160 $ superior blend packages.

This combination will maximize the profit, yielding a total profit of \$1,072 for the week.

Would you like to go through the steps in more detail, or do you have any specific questions?

Here are 5 related questions for further exploration:
1. How would the result change if the profit per superior blend package increased to \$5?
2. What if the availability of A grade coffee was reduced to 1,500 ounces?
3. How does the coffee house's maximum profit change if both blends have the same profit per package?
4. Could we solve this using graphical methods? How would the graph look?
5. How would the solution change if more filler ingredients were used, reducing the amount of coffee per blend?

**Tip:** Linear programming problems can be visualized by graphing the constraints and finding the feasible region, where the objective function is optimized.

For a given week, Abdul's Coffee House has available 1728 ounces of A grade coffee and 1760 ounces of B grade coffee. These are blended into 1-pound packages as follows: an economy blend that contains 2 ounces of A grade coffee and 10 ounces of B grade coffee, and a superior blend that contains 9 ounces of A grade coffee and 2 ounces of B grade coffee. There is a $3 profit on each economy blend package sold and a $4 profit on each superior blend package sold. Assuming that the coffee house is able to sell as many blends as it makes, how many packages of each blend should it make to maximize its profit for the week?

Learn how to solve a linear programming problem involving the production of coffee blends with limited coffee resources. This problem involves maximizing profit for Abdul's Coffee House by balancing two coffee blends (economy and superior) under constraints. Suitable for college students or advanced high school students interested in optimization techniques.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation