Longest backyard bisecting line

To find the longest straight line that can bisect Bob and Alice's backyard, we need to determine the line that divides the rectangular area into two equal halves. This line will be the longest straight line that bisects the yard.

Since the backyard is rectangular and its area is 2500 square feet, we can assume that the yard has sides of length a and b, where a × b = 2500.

Now, let's find the values of a and b. To do this, we can set up an equation based on the given information. Let's assume the length of the yard (a) is greater than or equal to the width (b):

a × b = 2500 ...........(1)

We also know that Alice walks 50 feet from her back door to the neighbor. Since the longest straight line that bisects the yard will pass through the midpoint of the yard, it will divide the length a into two equal halves, each measuring a/2. The line will also connect the back door with the neighbor's point on the opposite side. This creates a right-angled triangle, where the length of one side is a/2, the length of the other side is b, and the hypotenuse is 50 feet.

Using the Pythagorean theorem, we have:

(a/2)^2 + b^2 = 50^2 a^2/4 + b^2 = 2500 ...........(2)

Now, we can solve equations (1) and (2) simultaneously to find the values of a and b.

From equation (1): b = 2500/a

Substitute this value into equation (2):

a^2/4 + (2500/a)^2 = 2500 a^2/4 + 6250000/a^2 = 2500

Multiply everything by a^2 to get rid of the fraction:

a^4/4 + 6250000 = 2500a^2

Now, let's make a substitution: Let x = a^2:

x^2/4 + 6250000 = 2500x

Rearrange the equation:

x^2 - 10000x + 6250000 = 0

Now, we can solve this quadratic equation to find the value of x (which is a^2) using the quadratic formula:

x = [10000 ± √(10000^2 - 4 * 6250000)] / 2 x = [10000 ± √(100000000 - 25000000)] / 2 x = [10000 ± √75000000] / 2 x = [10000 ± 5000√3] / 2

Now we have two possible values for a^2:

1. a^2 = [10000 + 5000√3] / 2
2. a^2 = [10000 - 5000√3] / 2

Since the length cannot be negative, we discard the second solution. So, a^2 = [10000 + 5000√3] / 2.

Now, we can find the value of a:

a = √([10000 + 5000√3] / 2)

With a known value of a, we can find the corresponding value of b:

b = 2500 / a

Now, we have the lengths of the sides of the backyard (a and b). To find the longest straight line that bisects the yard, we take the diagonal of the rectangle, which is the hypotenuse of the right-angled triangle formed by sides a/2 and b:

Longest straight line = √((a/2)^2 + b^2)

Substitute the values of a and b to get the answer.