Elementary Mathematics from an Advanced Standpoint
Homework Sample
The Colorful Midwest
- Determine which, if any, of the color schemes
below satisfy the first two constrains. If a color scheme doesn’t
work, tell why.
- This scheme doesn’t work because it does not satisfy constrain number one, which states that no two states that share a border can be painted the same color. On the other hand, it does satisfy the second constrain. Kentucky and Illinois are two states that share a border and are painted with the color yellow, as well, South Dakota and Iowa share the color green.
- This scheme does satisfy both constrains, because none of the boarders are painted with the same color and each state is painted a shingle solid color.
- Below are two legal color schemes. Compute the total cost of implementing each of the two color schemes.
|
Question #2 a. |
|
Question #2 b. |
||||||
|
State |
Square mi |
Price |
Price per mi^2 |
|
State |
Square mi |
Price |
Price per mi^2 |
|
Illinois |
66,000 |
$2,000.00 |
$132,000,000.00 |
|
Illinois |
66,000 |
$1,500.00 |
$99,000,000.00 |
|
Indiana |
36,000 |
$1,000.00 |
$36,000,000.00 |
|
Indiana |
36,000 |
$3,000.00 |
$108,000,000.00 |
|
Iowa |
56,000 |
$1,000.00 |
$56,000,000.00 |
|
Iowa |
56,000 |
$3,000.00 |
$168,000,000.00 |
|
Kansas |
82,000 |
$3,000.00 |
$246,000,000.00 |
|
Kansas |
82,000 |
$1,500.00 |
$123,000,000.00 |
|
Kentucky |
40,000 |
$3,000.00 |
$120,000,000.00 |
|
Kentucky |
40,000 |
$2,000.00 |
$80,000,000.00 |
|
Michigan |
57,000 |
$1,500.00 |
$85,500,000.00 |
|
Michigan |
57,000 |
$2,000.00 |
$114,000,000.00 |
|
Minnesota |
80,000 |
$2,000.00 |
$160,000,000.00 |
|
Minnesota |
80,000 |
$1,500.00 |
$120,000,000.00 |
|
Missouri |
69,000 |
$1,500.00 |
$103,500,000.00 |
|
Missouri |
69,000 |
$1,000.00 |
$69,000,000.00 |
|
Nebraska |
77,000 |
$2,000.00 |
$154,000,000.00 |
|
Nebraska |
77,000 |
$2,000.00 |
$154,000,000.00 |
|
North Dakota |
70,000 |
$1,500.00 |
$105,000,000.00 |
|
North Dakota |
70,000 |
$2,000.00 |
$140,000,000.00 |
|
Ohio |
48,000 |
$2,000.00 |
$96,000,000.00 |
|
Ohio |
48,000 |
$1,500.00 |
$72,000,000.00 |
|
South Dakota |
87,000 |
$3,000.00 |
$261,000,000.00 |
|
South Dakota |
87,000 |
$1,000.00 |
$87,000,000.00 |
|
Wisconsin |
55,000 |
$3,000.00 |
$165,000,000.00 |
|
Wisconsin |
55,000 |
$1,000.00 |
$55,000,000.00 |
|
|
|
Total |
$1,720,000,000.00 |
|
|
|
Total |
$1,389,000,000.00 |
- If we ignored the constraint about states sharing common borders being different colors, then how many different combinations of colors could we come up with? 715 ways.
13C 4 = 13! = 13! = 715
4!(13-4)! 4! (9!)
- Come up with the cheapest legal color scheme that you can, and color the map below (or a copy of it) using that scheme. Be sure that you tell me what the total cost will be, how you came up with that scheme, and why you think it is going to be better than any other color scheme that anyone else may come up with. As usual, this is the real meat, and I expect much more than just a simple answer.
In order to find the answer for this question, I did a lot of combinations, but since we want to minimize the cost, this was the one that gave me the cheapest price. The way that I came up with this answer was by giving the lowest price to the states with the biggest square miles. For example, South Dakota, Kansas and Illinois were painted Red which only cost $1000 a square mile. Illinois is not one of the biggest states, but because we have to get this answer with the legal color scheme, it was chosen. My idea was to minimized the square miles for the most expensive color that was green and from there I continue. Below is the chart with the cost and a map colored.
|
State |
Square mile |
Price |
Price per square mile |
|
Illinois |
66,000 |
$1,000.00 |
$66,000,000.00 |
|
Indiana |
36,000 |
$2,000.00 |
$72,000,000.00 |
|
Iowa |
56,000 |
$3,000.00 |
$168,000,000.00 |
|
Kansas |
82,000 |
$1,000.00 |
$82,000,000.00 |
|
Kentucky |
40,000 |
$3,000.00 |
$120,000,000.00 |
|
Michigan |
57,000 |
$1,500.00 |
$85,500,000.00 |
|
Minnesota |
80,000 |
$1,500.00 |
$120,000,000.00 |
|
Missouri |
69,000 |
$2,000.00 |
$138,000,000.00 |
|
Nebraska |
77,000 |
$1,500.00 |
$115,500,000.00 |
|
North Dakota |
70,000 |
$2,000.00 |
$140,000,000.00 |
|
Ohio |
48,000 |
$1,000.00 |
$48,000,000.00 |
|
South Dakota |
87,000 |
$1,000.00 |
$87,000,000.00 |
|
Wisconsin |
55,000 |
$2,000.00 |
$110,000,000.00 |
|
|
|
Total |
$1,352,000,000.00 |
