ou have $98,000 you would like to invest in three different stocks: MarkyB Inc., JohnJohn Ltd., and Garretts Spaghetti House. You would like to invest no more than $18,000 in Garretts Spaghetti House, and you want the amount you invest in MarkyB Inc., to be at least the amount invested in JohnJohn Ltd. and Garretts Spaghetti House combined.
If MarkyB Inc. earns 18% annual interest, JohnJohn Ltd. earns 15% annual interest, and Garretts Spaghetti House earns 7% annual interest, how much money (in dollars) should you invest in each stock to maximize your annual interest earned? What is the maximum annual interest earned?

This is an optimization problem where we want to maximize the total annual interest earned from investments in three different stocks while satisfying specific constraints.

### Variables:
- Let $ x_1 $ represent the amount of money invested in **MarkyB Inc.**.
- Let $ x_2 $ represent the amount of money invested in **JohnJohn Ltd.**.
- Let $ x_3 $ represent the amount of money invested in **Garretts Spaghetti House**.

### Objective:
We want to maximize the total interest earned, which is given by:
\[
\text{Total interest earned} = 0.18x_1 + 0.15x_2 + 0.07x_3
\]
This is our **objective function**.

### Constraints:
1. The total amount invested is $98,000:
   \[
   x_1 + x_2 + x_3 = 98,000
   \]
2. No more than $18,000 is invested in Garretts Spaghetti House:
   \[
   x_3 \leq 18,000
   \]
3. The amount invested in MarkyB Inc. should be at least the sum of the amounts invested in JohnJohn Ltd. and Garretts Spaghetti House:
   \[
   x_1 \geq x_2 + x_3
   \]
4. All investments must be non-negative:
   \[
   x_1 \geq 0, \quad x_2 \geq 0, \quad x_3 \geq 0
   \]

### Step-by-step solution:
We'll approach this by solving the linear programming problem, which maximizes the interest function subject to the constraints.

#### 1. Write the problem in standard form:

Maximize:
\[
0.18x_1 + 0.15x_2 + 0.07x_3
\]

Subject to:
\[
x_1 + x_2 + x_3 = 98,000
\]
\[
x_3 \leq 18,000
\]
\[
x_1 \geq x_2 + x_3
\]
\[
x_1 \geq 0, \quad x_2 \geq 0, \quad x_3 \geq 0
\]

#### 2. Solve the system (I will calculate the optimal values for $x_1$, $x_2$, and $x_3$).

Let me compute this.The optimal investment strategy is as follows:
- Invest **$98,000** in **MarkyB Inc.**.
- Invest **$0** in **JohnJohn Ltd.**.
- Invest **$0** in **Garretts Spaghetti House**.

The maximum annual interest earned is **$17,640**.

This solution satisfies the constraints, including that MarkyB Inc. earns the highest interest rate (18%), so the best strategy is to invest the entire amount there.

Would you like more details, or do you have any questions?

### Related Questions:
1. What happens if the constraint $x_1 \geq x_2 + x_3$ is removed?
2. What if you had to invest a minimum amount in Garretts Spaghetti House?
3. How would the solution change if JohnJohn Ltd. had a higher interest rate?
4. Can we invest an equal amount in all three companies and still maximize the interest?
5. What is the impact of changing the interest rates on the optimal investment?

### Tip:
When solving optimization problems, always check if one asset has significantly better returns, as it may dominate the solution under certain constraints.

You have $98,000 you would like to invest in three different stocks: MarkyB Inc., JohnJohn Ltd., and Garretts Spaghetti House. You would like to invest no more than $18,000 in Garretts Spaghetti House, and you want the amount you invest in MarkyB Inc. to be at least the amount invested in JohnJohn Ltd. and Garretts Spaghetti House combined. If MarkyB Inc. earns 18% annual interest, JohnJohn Ltd. earns 15% annual interest, and Garretts Spaghetti House earns 7% annual interest, how much money (in dollars) should you invest in each stock to maximize your annual interest earned? What is the maximum annual interest earned?

Discover how to maximize annual interest earned from an investment of $98,000 across MarkyB Inc., JohnJohn Ltd., and Garretts Spaghetti House. Learn key linear programming techniques and optimization strategies to allocate funds under specific constraints, earning a maximum of $17,640 annually.

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