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. In this case, the net profit can be calculated by multiplying the number of graphing calculators (x) by the profit per graphing calculator ($5), and subtracting the number of scientific calculators (y) multiplied by the loss per scientific calculator ($2).

Objective equation: Net Profit = 5x - 2y

B) The constraint equations can be derived from the given information:

1. Demand Constraint:
The demand for scientific calculators each day is at least 100, and the demand for graphing calculators is at least 80.

Scientific Calculator Constraint: y ≥ 100
Graphing Calculator Constraint: x ≥ 80

2. Production Capacity Constraint:
The production capacity for scientific calculators is a maximum of 200, and for graphing calculators is a maximum of 170.

Scientific Calculator Constraint: y ≤ 200
Graphing Calculator Constraint: x ≤ 170

3. Shipping Contract Constraint:
A total of at least 200 calculators must be shipped each day.

Total Calculator Constraint: x + y ≥ 200

C) To graph the associated constraint equations, we can plot them on a Cartesian plane.

D) To determine the vertices of each of the graphed equations, we need to find the points where the different constraints intersect. These intersection points are the vertices.

E) To determine the optimal number of each type of calculator to maximize net profits, we need to find the vertex that results in the highest net profit value.

We will find the solution in the following steps.

A) Objective Equation:

To maximize net profits, we need to define the objective equation. Let's assume x represents the number of scientific calculators produced daily, and y represents the number of graphing calculators produced daily.

Profit from scientific calculators = profit per scientific calculator * number of scientific calculators
Profit from graphing calculators = profit per graphing calculator * number of graphing calculators

Objective Equation: P = -2x + 5y

B) Constraint Equations:
We have the following constraints based on the given information:
1) Demand constraint:
x ≥ 100 (At least 100 scientific calculators daily)
y ≥ 80 (At least 80 graphing calculators daily)

2) Production capacity constraint:
x ≤ 200 (No more than 200 scientific calculators daily)
y ≤ 170 (No more than 170 graphing calculators daily)

3) Shipping contract constraint:
x + y ≥ 200 (At least 200 calculators must be shipped daily)

C) Graphing the associated constraint equations:
To visualize the constraint equations, we plot the feasible region on a graph. The feasible region represents the area where all the constraints are satisfied.

D) Determining the vertices of each of the graphed equations:
The vertices of the feasible region are the points where the constraint lines intersect. We find the coordinates of these vertices by solving the equations simultaneously.

E) Determining the optimal number of calculators and net profit:
To determine the optimal number of each type of calculator and the corresponding net profit, we substitute the coordinates of the feasible region's vertices into the objective equation and calculate the net profit. The solution with the highest net profit will give us the answer.