A contractor builds two types of homes. The Carolina requires one lot, $160,000 capital, and 160 worker-days of labor, whereas the Savannah requires one lot, $240,000 capital, and 160 worker-days of labor. The contractor owns 300 lots and has $48,000,000 available capital and 43,200 worker-days of labor. The profit on the Carolina is $40,000 and the profit on the Savannah is $52,000. Use the corner points of the feasible region to find how many of each type of home should be built to maximize profit. Find the maximum possible profit. Round your profit to the nearest dollar, another answers - to the nearest whole number.
A)Maximum profit of $13,320,000 obtained by building 60 Carolina homes and 210 Savannah homes.
B)Maximum profit of $12,000,000 obtained by building 300 Carolina homes and 0 Savannah homes.
C)Maximum profit of $11,520,000 obtained by building 210 Carolina homes and 60 Savannah homes.
D)Maximum profit of $21,200,000 obtained by building 270 Carolina homes and 200 Savannah homes.
E)Maximum profit of $18,400,000 obtained by building 200 Carolina homes and 200 Savannah homes.

Question

A contractor builds two types of homes. The Carolina requires one lot, $160,000 capital, and 160 worker-days of labor, whereas the Savannah requires one lot, $240,000 capital, and 160 worker-days of labor. The contractor owns 300 lots and has $48,000,000 available capital and 43,200 worker-days of labor. The profit on the Carolina is $40,000 and the profit on the Savannah is $52,000. Use the corner points of the feasible region to find how many of each type of home should be built to maximize profit. Find the maximum possible profit. Round your profit to the nearest dollar, another answers - to the nearest whole number. ​
A)Maximum profit of $13,320,000 obtained by building 60 Carolina homes and 210 Savannah homes.
B)Maximum profit of $12,000,000 obtained by building 300 Carolina homes and 0 Savannah homes.
C)Maximum profit of $11,520,000 obtained by building 210 Carolina homes and 60 Savannah homes.
D)Maximum profit of $21,200,000 obtained by building 270 Carolina homes and 200 Savannah homes.
E)Maximum profit of $18,400,000 obtained by building 200 Carolina homes and 200 Savannah homes.

Quizplus · Accepted Answer

To solve this problem, we need to determine the corner points of the feasible region, which represents the limits on the number of homes that can be built given the available resources. The corner points are found by setting each of the constraints equal to zero and solving for the variables.

Corner point A: 0 Carolina homes, 0 Savannah homes
Capital constraint: 0 + 0 ≤ 48,000,000
Labor constraint: 0 + 0 ≤ 43,200
Feasible

Corner point B: 0 Carolina homes, 300 Savannah homes
Capital constraint: 0 + (300 x $240,000) ≤ 48,000,000
Labor constraint: 0 + (300 x 160) ≤ 43,200
Infeasible

Corner point C: 210 Carolina homes, 60 Savannah homes
Capital constraint: (210 x $160,000) + (60 x $240,000) ≤ 48,000,000
Labor constraint: (210 x 160) + (60 x 160) ≤ 43,200
Feasible

Corner point D: 270 Carolina homes, 200 Savannah homes
Capital constraint: (270 x $160,000) + (200 x $240,000) ≤ 48,000,000
Labor constraint: (270 x 160) + (200 x 160) ≤ 43,200
Infeasible

Corner point E: 200 Carolina homes, 200 Savannah homes
Capital constraint: (200 x $160,000) + (200 x $240,000) ≤ 48,000,000
Labor constraint: (200 x 160) + (200 x 160) ≤ 43,200
Feasible

Next, we need to calculate the profit for each corner point by multiplying the number of homes of each type by its respective profit, and then summing the products.

Corner point A: 0 ($0) + 0 ($0) = $0
Corner point C: 210 ($40,000) + 60 ($52,000) = $11,520,000
Corner point E: 200 ($40,000) + 200 ($52,000) = $18,400,000

Since corner point E has the highest profit, the best choice is to build 200 Carolina homes and 200 Savannah homes, for a maximum profit of $18,400,000.

A Contractor Builds Two Types of Homes