Final answer:
The shortest ladder to clear an 8-foot-tall fence 3 feet away from a house is found using the Pythagorean theorem. After calculations, it is approximately 8.54 feet long.
Step-by-step explanation:
The question asks for the length of the shortest ladder that can clear an 8-foot-tall fence and reach a house 3 feet away. This is a classic application of the Pythagorean theorem in mathematics, which 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 lengths of the other two sides. Here, the fence and the distance from the fence to the house form the two shorter sides of the triangle, and the ladder will be the hypotenuse.
To find the shortest ladder length that can clear the fence, we'll use the Pythagorean theorem:
Ladder length2 = Fence height2 + Distance from house2. Converting the height of the fence to feet (8 feet) and keeping the distance from the house as it is (3 feet), we calculate:
Ladder length2 = 82 + 32 = 64 + 9 = 73
Therefore, the ladder length = √73 ≈ 8.54 feet. The shortest ladder that can clear the fence and reach the house is approximately 8.54 feet long.