Shortest ladder A 10-ft-tall fence runs parallel to the wall of a house at a distance of 4 ft. Find the length of the shortest ladder that extends from the ground to the house without touching the fence. Assume the vertical wall of the house and the horizontal ground have infinite extent.

Verified step by step guidance

1. Visualize the problem by sketching the scenario: draw the house wall, the fence, and the ground, marking the distances involved (4 ft from the wall to the fence and the height of the fence at 10 ft).
2. Define the variables: let 'L' be the length of the ladder, 'x' be the horizontal distance from the base of the ladder to the wall of the house, and 'y' be the height at which the ladder touches the wall.
3. Use the Pythagorean theorem to express the length of the ladder in terms of 'x' and 'y': L = √(x² + y²).
4. Set up a relationship between 'x' and 'y' using similar triangles, considering the triangle formed by the ladder, the wall, and the ground, and the triangle formed by the fence and the ground.
5. Minimize the length 'L' by taking the derivative of the expression obtained in step 3 with respect to 'x', setting it to zero to find critical points, and determining the minimum length of the ladder.