The Farmer's Planting Dilemma.

[TchrWill]

A farmer wishes to lay out 3 rectangular planting areas having integer dimensions. The 3 rectangles have identical areas. The first rectangle is 278 feet longer than it is wide. The second rectangle is 96 feet longer than it is wide. The third rectangle is 542 feet longer than it is wide. Find the dimensions and area of the rectangles.

 

[Denis]

1164 by 1260, 1080 by 1358, 970 by 1512 ; areas = 1,466,640
Starting point: widths = a,b,c with a>b>c; then:
[96^2 + 4b^2 + 4(278b)] and [96^2 + 4c^2 + 4(542c)] must both be squares.

 

[Marishipu]

1164 by 1260, 1080 by 1358, 970 by 1512 ; areas = 1,466,640
Starting point: widths = a,b,c with a>b>c; then:
[96^2 + 4b^2 + 4(278b)] and [96^2 + 4c^2 + 4(542c)] must both be squares.

Would you please expand on this solution (i.e., show how it is actually solved)

 

[Marishipu]

A farmer wishes to lay out 3 rectangular planting areas having integer dimensions. The 3 rectangles have identical areas. The first rectangle is 278 feet longer than it is wide. The second rectangle is 96 feet longer than it is wide. The third rectangle is 542 feet longer than it is wide. Find the dimensions and area of the rectangles.

For this solution:
1164 by 1260, 1080 by 1358, 970 by 1512 ; areas = 1,466,640
Starting point: widths = a,b,c with a>b>c; then:
[96^2 + 4b^2 + 4(278b)] and [96^2 + 4c^2 + 4(542c)] must both be squares.

Would you please expand on this solution (i.e., show how it is actually solved)

 

[Deleted member 4993]

For this solution:
1164 by 1260, 1080 by 1358, 970 by 1512 ; areas = 1,466,640
Starting point: widths = a,b,c with a>b>c; then:
[96^2 + 4b^2 + 4(278b)] and [96^2 + 4c^2 + 4(542c)] must both be squares.

Would you please expand on this solution (i.e., show how it is actually solved)

What have you tried?

Please share your attempt/s ?

 

[HallsofIvy]

A farmer wishes to lay out 3 rectangular planting areas having integer dimensions. The 3 rectangles have identical areas. The first rectangle is 278 feet longer than it is wide. The second rectangle is 96 feet longer than it is wide. The third rectangle is 542 feet longer than it is wide. Find the dimensions and area of the rectangles.

So the first rectangle is i feet wide and i+ 278 feet long. Its area is i(i+ 278).
The second rectangle is j feet wide and j+ 96 feet long. Its area is j(j+ 96).
The third rectangle is k feet wide and k+ 542 feet long. Its area is k(k+ 542).

Since they all have the same area we have the two equations
i(i+ 278)= j(j+ 96) and
j(j+ 96)= k(k+ 542).

[i(i+ 278)= k(k+ 542) is also true but is not an independent equation.]

And, of course, i, j, and k must be integers.

 

[mmm4444bot]

Would you please expand on this solution (i.e., show how it is actually solved)

I think Denis had written a computer program to find a,b,c values that produced perfect squares in his expressions. In other words, he'd used an automated, guess-and-check approach.

?

 

[Marishipu]

I built a program in MS Excel Solver that uses brute force and guessing.
I was wondering if there is a way to solve it mathematically.

 

[Marishipu]

I was wondering if there is a way to solve it mathematically.

 

[blamocur]

I was wondering if there is a way to solve it mathematically.

There is a way to both solve it and to show that the solution is unique. My approach starts with replacing the variables so that each area can be expressed as a difference of squares. Let me know if you need more hints.

 

[Marishipu]

There is a way to both solve it and to show that the solution is unique. My approach starts with replacing the variables so that each area can be expressed as a difference of squares. Let me know if you need more hints.

Thank you for responding. Yes, I do not know how to do this. Showing the math would be most helpful.

 

[blamocur]

Thank you for responding. Yes, I do not know how to do this. Showing the math would be most helpful.

For example, in the first case the area [imath]S=x(x+278)[/imath], but if one uses [imath]u=x+139[/imath] then [imath]S=u^2-139^2[/imath]. Similar transformations of the remaining two expressions allow us to rewrite the equality. After that one can exploit the fact that all the variables must be integers.