A fence 2 feet tall runs parallel to a tall building at a distance of 6 feet from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building?

Let the ladder length be d

Let the base of the ladder be x feet from the fence, and hit the wall at height h.

from similar triangles,

2/x = h/(6+x)
h = 2(x+6)/x

d^2 = (6+x)^2 + h^2
= (x+6)^2 + 4(x+6)^2/x^2
= (x+6)^2 * (1 + 4/x^2)

d^2 (and therefore d) has a minimum at x = 2.9

d^2 = 116.88
d = 10.8 ft

To determine the length of the shortest ladder that will reach from the ground over the fence to the wall of the building, we can use the Pythagorean theorem.

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

In this scenario, the ground, the fence, and the wall of the building form a right triangle. The height of the fence is one side, the distance between the fence and the building is the other side, and the length of the ladder is the hypotenuse.

Since the height of the fence is given as 2 feet, and the distance between the fence and the building is given as 6 feet, we can use these values in the Pythagorean theorem formula.

To calculate the length of the ladder, we square the height of the fence and the distance between the fence and the building, then sum the two values and take the square root of the result:

Ladder Length = sqrt((height of fence)^2 + (distance to building)^2)
Ladder Length = sqrt(2^2 + 6^2)
Ladder Length = sqrt(4 + 36)
Ladder Length = sqrt(40)
Ladder Length ≈ 6.32 feet

Therefore, the length of the shortest ladder that will reach from the ground over the fence to the wall of the building is approximately 6.32 feet.