Answer
0 colonial, 90 ranch
Some lots (10) will be left vacant.
Work Step by Step
Define x as the number of colonial houses, and y as the number of ranch houses. Since there is 4000 dollars of profit per colonial house, and 8000 dollars of profit per ranch house, we want to maximize f(x) = 4000x + 8000y.
The farm only has room for 100 houses, so x + y $\leq$ 100. Each colonial house costs 30,000 dollars and each ranch house costs 40,000 dollars, so the total cost is 30000x + 40000y. The contractor also only has 3.6 million dollars, so 30000x + 40000y $\leq$ 3600000. We also need to have x $\geq$ 0 and y $\geq$ 0, since we can't have negative houses. Now, we can graph these constraints:
Since maximums occur at the corners of the area, we take the coordinates of the corners: (0,0), (0,90), (40,60), and (100,0). Now, we plug in the x and y values into f(x):
(0,0): f(x) = 4000(0) + 8000(0) = 0
(0,90): f(x) = 4000(0) + 8000(90) = 720000
(40,60): f(x) = 4000(40) + 8000(60) = 160000+480000 = 640000
(100,0): f(x) = 4000(100) + 8000(0) = 400000
Since the maximum occurs when x = 0 and y = 90, the contractor should build 90 ranch houses and no colonial houses. And some lots will be left blank (10 to be exact).
