Longest diagonal not found.

To find the longest straight line that can bisect Bob and Alice's backyard, we need to find the diagonal of the rectangular backyard. The diagonal of a rectangle is the longest straight line that connects opposite corners.

Let's assume the length of the rectangular backyard is 'L' and the width is 'W'. The area of the rectangle is given as 28211 square feet, so we have:

Area = Length (L) * Width (W) = 28211 square feet

We also know that every morning, Alice walks 507 feet from her back door to the neighbor, which is essentially the diagonal of the rectangle.

Now, we can use the Pythagorean theorem to find the length of the diagonal (d) of the rectangle:

d^2 = L^2 + W^2

But we still need the values of L and W to proceed. To find these values, we can use the fact that the diagonal is 507 feet long and the area is 28211 square feet:

Let's find the possible values of L and W by considering various combinations of factors of the area 28211:

1. Area = L * W = 28211
2. L^2 + W^2 = 507^2 (since the diagonal is 507 feet)

Now, let's find the factors of 28211 and check for possible combinations that satisfy the Pythagorean equation:

Factors of 28211: 1, 11, 2561, 28211

Now, let's check the combinations:

1. L = 1, W = 28211: 1^2 + 28211^2 = 795160521 (not equal to 507^2)
2. L = 11, W = 2561: 11^2 + 2561^2 = 6554302 (not equal to 507^2)
3. L = 2561, W = 11: 2561^2 + 11^2 = 6554302 (not equal to 507^2)
4. L = 28211, W = 1: 28211^2 + 1^2 = 795160521 (not equal to 507^2)

It seems that none of these combinations satisfy the Pythagorean equation with the diagonal being 507 feet.

Since there is no exact solution for this specific scenario, we might be dealing with some rounding errors or approximations in the given information. To ensure a correct answer, we may need to reevaluate the problem or verify the provided data.