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

Answer 1

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.