A manufacturer produces two different types of candy - hard candy and gummies. The hard candies contribute $50 per box sold and the gummies $30 per box towards the profit. Raw materials of sugar and food coloring are required for production. At least 18 kg of sugar and 12 kg of food coloring must be used daily. Also, at most 34 hours of labor are to be utilized. A quantity of 2kg of sugar is needed for hard candies and 1 kg of sugar for gummies. 1 kg of food coloring is needed for both hard candy and gummies. It takes 2 hours to manufacture hard candies and 2 hours to manufacturer gummies. How many units of each model should be produced in order to maximize the profit? A. Write the constraints B. Write the objective function C. Graph the constraints. Shade in feasible region D. Find and test the points that will probably maximize the objective function E. Write the answer as a a complete sentence
A. Constraints:
Let x be the number of boxes of hard candy to produce
Let y be the number of boxes of gummies to produce
Sugar constraint: 2x + y ≤ 18
Food coloring constraint: x + y ≤ 12
Labor constraint: 2x + 2y ≤ 34
Non-negativity constraint: x ≥ 0, y ≥ 0
B. Objective function:
Let P be the profit
P = 50x + 30y
C. Graphing the constraints:
The feasible region is the area where all the constraints are satisfied. To graph the constraints, we plot the boundary lines and shade the region that satisfies all the constraints.
For the sugar constraint, when 2x + y = 18:
x-intercept: x = 9
y-intercept: y = 18
Plotting the line gives us a straight line passing through (9, 0) and (0, 18).
For the food coloring constraint, when x + y = 12:
x-intercept: x = 12
y-intercept: y = 12
Plotting the line gives us a straight line passing through (12, 0) and (0, 12).
For the labor constraint, when 2x + 2y = 34:
x-intercept: x = 17
y-intercept: y = 17
Plotting the line gives us a straight line passing through (17, 0) and (0, 17).
Shading in the feasible region includes the area below or on the lines of the constraints.
D. Finding and testing the points that will probably maximize the objective function:
To find the points that will maximize the objective function, we need to find the corner points of the feasible region.
The corner points are the intersection points of the boundary lines. The corner points are: (0, 12), (0, 18), (9, 0), and (6, 6).
To test each corner point, substitute the x and y values into the objective function P = 50x + 30y.
At (0, 12), P = 50(0) + 30(12) = 360
At (0, 18), P = 50(0) + 30(18) = 540
At (9, 0), P = 50(9) + 30(0) = 450
At (6, 6), P = 50(6) + 30(6) = 480
E. Writing the answer as a complete sentence:
The manufacturer should produce 0 boxes of hard candies and 18 boxes of gummies to maximize profit.
A. The constraints for this problem can be formulated as follows:
1. Sugar constraint: 2x + y ≥ 18 (where x represents the number of boxes of hard candies and y represents the number of boxes of gummies)
2. Food coloring constraint: x + y ≥ 12
3. Labor constraint: 2x + 2y ≤ 34 (since it takes 2 hours to manufacture both hard candies and gummies)
B. The objective function for this problem is to maximize the profit, which can be represented as:
Profit = 50x + 30y (where x represents the number of boxes of hard candies and y represents the number of boxes of gummies)
C. Let's graph the constraints using a coordinate system:
The sugar constraint: 2x + y ≥ 18
- Convert the inequality to an equation: 2x + y = 18
- Plot the line 2x + y = 18
- Shade the region above or on the line since it is a greater than or equal to constraint
The food coloring constraint: x + y ≥ 12
- Convert the inequality to an equation: x + y = 12
- Plot the line x + y = 12
- Shade the region above or on the line since it is a greater than or equal to constraint
The labor constraint: 2x + 2y ≤ 34
- Convert the inequality to an equation: 2x + 2y = 34
- Divide the equation by 2 to simplify: x + y = 17
- Plot the line x + y = 17
- Shade the region below or on the line since it is a less than or equal to constraint
D. To find the points that will probably maximize the objective function (50x + 30y), we need to find the corner points of the feasible region (the shaded area where all constraints are satisfied).
- The corner points can be found by finding the intersection points of the lines formed by the constraints.
E. Without the specific graph, the exact corner points and feasible region cannot be determined. After determining the corner points, the objective function would be tested at each corner point to find the maximum profit. The answer would be written as a complete sentence once the corner points and objective function values are known.