A company is planning to purchase and store two items, gadgets and widgets. Each gadget costs $2.00 and occupies 2 square meters of floor space; each widget costs $3.00 and occupies 1 square meter of floor space. $1,200 is available for purchasing these items and 800 square meters of floor space is available to store them. Each gadget contributes $3.00 to profit and each widget contributes $2.00 to profit.
Identify all constraints.
Identify all applicable corner points of the feasibility region.
What combination of gadgets and widgets produces maximum profit?
To identify the constraints, we need to consider the limitations of budget and floor space.
1. Budget constraint: The total cost of gadgets and widgets must not exceed the available budget of $1,200. The cost of gadgets is $2.00 per unit, and the cost of widgets is $3.00 per unit.
2. Floor space constraint: The total floor space occupied by gadgets and widgets must not exceed the available floor space of 800 square meters. Gadgets occupy 2 square meters each, and widgets occupy 1 square meter each.
Let's find the corner points of the feasibility region where both constraints are satisfied.
To do this, we can set up a system of inequalities:
Let G = the number of gadgets
Let W = the number of widgets
Budget constraint: 2G + 3W ≤ 1,200
Floor space constraint: 2G + W ≤ 800
To find the feasible region, we need to solve this system of inequalities. However, it is important to note that as this questions asks for feasible corner points only, substitution method will be used to find the coordinates of the corner points. First youb set the equations equal to one another.
-2G - W = -400
For the sake of simplicity, we'll assume G and W are whole numbers.
Now, we need to find the feasible values for G and W.
Constraint 1: 2G + 3W ≤ 1,200
- Let's assume G = 0, then we have 3W ≤ 1,200
- W can be 0, 1, 2,..., 400, but the cost of widgets must also be feasible within the given budget constraint.
- If G = 1, we have 2 + 3W ≤ 1,200
- W can be 0, 1, 2,..., 399
- Continuing this pattern, we can find feasible values for higher G as well.
Similarly, we can find the feasible values for G and W based on constraint 2: 2G + W ≤ 800.
Once we have all the feasible values for G and W, we can plot them on a graph and identify the corner points of the feasible region.
To find the combination of gadgets and widgets that produces maximum profit, we need to consider the profit contributed by each gadget and widget.
Profit per gadget = $3.00
Profit per widget = $2.00
To determine the maximum profit, we can calculate the profit for each feasible combination of gadgets and widgets and choose the one with the highest value.
By substituting the values of each feasible combination into the profit equation, we can determine the maximum profit.