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.

Question

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?

Answer 1

contraints

2 g + 3 w < 1200
2 g + w < 800
objective
p = 3 g + 2 w

corners (g,w)
(0,0)
if g = 0
w = 400 or 800, use 400 (0,400)
(0,400)
if w = 0
g = 600 or 400, use 400 (400,0)
intersection
2 g + 3 w = 1200
2 g + w = 800
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2 w = 400
w = 200
g = 300 so (300,200)
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look at p
at (0,0) is 0
at (0,400) is 800
at (400,0) is 1200
at (300,200) is 3*300+2*200 = 1300
max profit of 1300 at (300,200)

Answer 2

To identify the constraints in this scenario, we need to consider the limitations imposed by the available budget and floor space. Here are the constraints:

1. Budget constraint:
The total cost of purchasing gadgets (x) and widgets (y) should not exceed $1,200. This can be represented as: 2x + 3y ≤ 1200.

2. Floor space constraint:
The total floor space occupied by gadgets (x) and widgets (y) should not exceed 800 square meters. This can be represented as: 2x + y ≤ 800.

Now, let's find the corner points of the feasibility region by solving the system of inequalities formed by the constraints:

Step 1: Convert the inequalities to equations:
2x + 3y = 1200
2x + y = 800

Step 2: Solve the system of equations to find the corner point(s).

By solving the equations, we find two corner points:
Corner Point 1: (200, 400)
Corner Point 2: (400, 0)

Finally, to determine the combination of gadgets and widgets that produces the maximum profit, we need to calculate the profit for each corner point.

1. Corner Point 1: (200, 400)
Profit = (3 * 200) + (2 * 400) = $1,600

2. Corner Point 2: (400, 0)
Profit = (3 * 400) + (2 * 0) = $1,200

Hence, the combination of 200 gadgets and 400 widgets will produce the maximum profit of $1,600.