A candy shop puts together two prepackaged assortments to be given to​ trick-or-treaters on Halloween. The feasible set for producing x packages of assortment A and y packages of assortment B has been graphed to the right. For the earnings function Earningsequals40xplus30​y, where Earnings is in​ cents, how many of each assortment should be prepared in order to maximize​ profits? What is the maximum​ profit? The shop should prepare 
  
enter your response here packages of assortment A and 
  
enter your response here packages of assortment B to maximize profit.

To solve this problem, we need to maximize the shop's **earnings** given the **constraint** represented by the feasible set.

We are given:
- The **earnings function**: \( E = 40x + 30y \), where \( x \) is the number of packages of assortment A and \( y \) is the number of packages of assortment B.
- The feasible region for \( x \) and \( y \) (though not explicitly shown, it is described as graphed).

### Steps to Solve:

1. **Identify the feasible region**: The feasible set describes all possible combinations of \( x \) (packages of A) and \( y \) (packages of B) that the shop can produce, given certain constraints such as resources or time. It’s often bounded by lines representing those constraints.

2. **Determine the vertices of the feasible region**: The maximum or minimum value of the earnings function will occur at one of the vertices (corners) of the feasible region. You would evaluate the earnings function at each vertex to find where the maximum profit occurs.

3. **Evaluate the earnings function at the vertices**: For each vertex \((x_i, y_i)\), plug the values into the earnings function \( E(x, y) = 40x + 30y \).

4. **Find the maximum earnings**: Compare the results from each vertex and select the highest value.

Since the graph isn't provided here, I can guide you on how to proceed:

- If you have the coordinates of the vertices (which are the points where the constraints intersect), plug them into the earnings function \( E(x, y) = 40x + 30y \).
- The combination that gives the highest value for \( E \) is the number of packages \( x \) and \( y \) the shop should prepare to maximize profits.

Would you like me to assist you further in analyzing the graph or explain the process in more detail?

### Related Questions:
1. What is the significance of the feasible region in a linear programming problem?
2. How do you determine the vertices of a feasible set from a graph?
3. What are the main differences between maximization and minimization problems in linear programming?
4. How can constraints affect the number of packages the shop can prepare?
5. Can the shop increase profits by adjusting the coefficients in the earnings function?

### Tip:
Always check if the feasible region is **bounded** or **unbounded** when solving linear programming problems. A bounded region ensures a maximum or minimum solution.

A candy shop puts together two prepackaged assortments to be given to​ trick-or-treaters on Halloween. The feasible set for producing x packages of assortment A and y packages of assortment B has been graphed to the right. For the earnings function Earningsequals40xplus30​y, where Earnings is in​ cents, how many of each assortment should be prepared in order to maximize​ profits? What is the maximum​ profit? The shop should prepare enter your response here packages of assortment A and enter your response here packages of assortment B to maximize profit.

Learn how to maximize profits for a candy shop using linear programming. This problem involves finding the optimal number of two Halloween assortments to prepare, using the earnings function Earnings = 40x + 30y. Key concepts include optimization, the feasible region, and profit maximization. Suitable for Grades 9-12.

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