A fence 4 feet tall runs parallel to a tall building at a distance of 2 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?

Did you make a diagram?

Let the distance from the fence to the foot of the ladder be x feet.
Let the top of the ladder by y feet above the ground.
Let L be the length of the ladder.

L^2 = (x+2)^2 + y^2

by similar triangles:
4/x = y/(x+2)

so L^2 = (x+2)^2 + 16(x+2)^2/x^2
= (x+2)^2(1 + 16/x^2)

2L(dL/dx) = (x+2)^2(-32/x^3) + (1+16/x^2)(2)(x+2)
= (x+2)(-32(x+2)/x^3 + 2(1+16/x^2)

For a max/min of L, dL/dx = 0
so (x+2)(-32(x+2)/x^3 + 2(1+16/x^2) = 0
x = -2 clearly cannot be a solution , so (-32(x+2)/x^3 + 2(1+16/x^2) = 0
-32/x^2 - 64/x^3 + 2 + 32/x^2 = 0
2 = 64/x^3
x^3 = 32
x = 32^(1/3)

carefully sub x = 32^(1/3) back into the L^2 equation to find L.

(I got 8.32 feet)

To find 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-angled 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 other two sides.

In this case, the fence, the wall of the building, and the ground form a right-angled triangle. Let's label the sides of the triangle. The length of the fence is the adjacent side, the distance from the fence to the building is the opposite side, and the length of the ladder is the hypotenuse.

Given:
Height of the fence (adjacent side) = 4 feet
Distance from the fence to the building (opposite side) = 2 feet

Now, we can apply the Pythagorean theorem to find the length of the ladder:

hypotenuse^2 = adjacent^2 + opposite^2

ladder^2 = 4^2 + 2^2
ladder^2 = 16 + 4
ladder^2 = 20

To calculate the length of the ladder, we need to take the square root of both sides:

ladder = √20

Simplifying the square root of 20, we get:

ladder ≈ 4.47 feet

Therefore, the length of the shortest ladder required to reach from the ground over the fence to the wall of the building is approximately 4.47 feet.