A fence 3 feet tall runs parallel to a tall building at a distance of 3 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?

Draw a diagram. If the ladder's length is z, and the base of the ladder is x feet from the fence, then by similar triangles, you can see that

z/(x+3) = √(x^2+9)/x
z = (x+3)√(x^2+9)/x
dz/dx = (x^3-27) / (x^2 √(x^2+9))
dz/dx=0 when x=3

z(3) = 6√18/3 = 6√2

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 triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.

In this case, the ladder, the fence, and the distance from the fence to the building form a right triangle. The ladder is the hypotenuse, and the fence and the distance from the fence to the building are the other two sides.

Let's call the length of the ladder 'L'. The height of the fence is given as 3 feet, and the distance from the fence to the building is given as 3 feet.

Using the Pythagorean theorem, we can write the equation:

L^2 = 3^2 + 3^2

Simplifying the equation:

L^2 = 9 + 9
L^2 = 18

Taking the square root of both sides to find the length of the ladder:

L = √18

L ≈ 4.24 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 4.24 feet.