A calculator company produces a scientific calculator and a graphing calculator. Long-term projections indicate an expected demand of at least 100 scientific and 80 graphing calculators each day. Because of limitations on production capacity, no more than 200 scientific and 170 graphing calculators can be made daily. To satisfy a shipping contract, a total of at lease 200 calculators must be shipped each day.

If each scientific calculator sold results in a $2 loss,but each graphing calculator produces a $5 profit, how many if each type should be made daily to maximize net profits?

A) Define the objective equation.
B)Define the constraint equations.
C)Graph the associated constraint equations.
D) Determine the vertices of each of the graphed equations.
E) Determine how many of each type of calculators should be made daily to maximize the net profits and the amount of net profit.

A) The objective equation represents the net profit, which can be calculated by subtracting the cost from the revenue. In this case, the revenue is obtained from selling each graphing calculator, which produces a $5 profit, and the cost is associated with selling each scientific calculator, resulting in a $2 loss. Let's represent the number of scientific calculators with 'x' and the number of graphing calculators with 'y'. Therefore, the objective equation is:

Net Profit = 5y - 2x

B) The constraint equations represent the limitations and requirements given in the problem. The constraints are as follows:

1. At least 100 scientific calculators must be produced daily: x ≥ 100
2. At least 80 graphing calculators must be produced daily: y ≥ 80
3. No more than 200 scientific calculators can be made daily: x ≤ 200
4. No more than 170 graphing calculators can be made daily: y ≤ 170
5. A total of at least 200 calculators must be shipped daily: x + y ≥ 200

C) To graph the constraint equations, we need to plot them on a coordinate plane. Each equation will form a line, and the area of feasible solutions will be where these lines intersect or overlap.

D) Let's analyze each constraint equation separately to determine their respective vertices:

1. x ≥ 100:
This constraint equation represents the minimum requirement for scientific calculators. The line x = 100 is parallel to the y-axis.

2. y ≥ 80:
This constraint equation represents the minimum requirement for graphing calculators. The line y = 80 is parallel to the x-axis.

3. x ≤ 200:
This constraint equation represents the maximum capacity for scientific calculators. The line x = 200 is parallel to the y-axis.

4. y ≤ 170:
This constraint equation represents the maximum capacity for graphing calculators. The line y = 170 is parallel to the x-axis.

5. x + y ≥ 200:
This constraint equation represents the total number of calculators that must be shipped daily. To graph, convert it to slope-intercept form:
y ≥ 200 - x

E) To determine the optimal number of calculators and maximum net profit, we need to find the feasible region, which is the intersection of all the constraint lines. The vertices of this region will help us identify the extreme points.

After graphing the constraint equations and finding the points of intersection, we can determine the vertices. These vertices represent the different combinations of scientific and graphing calculators that satisfy all the constraints.

Once we have the vertices, we substitute the x and y values into the objective equation (Net Profit = 5y - 2x) to calculate the net profit at each vertex. The maximum net profit will correspond to the vertex with the highest net profit value.

To determine how many of each type of calculator should be made daily to maximize net profits and the amount of net profit, it would be necessary to complete the graphing, identify the vertices, calculate the net profits at each vertex, and compare the results.