Question

a tree is supported by a wire anchored in the ground 5 feet from its base. The wire is one foot longer than the height that it reaches on the tree. find the length of the wire

Answer 1

Task. A tree is supported by a wire anchored in the ground 5 feet from its base. The wire is one foot longer than the height that it reaches on the tree. Find the length of the wire.

Solution. Consider the figure:

Let $L$ be the length of the wire and $h$ be the height where it reaches the tree. Then by assumption

$$h = L - 1.$$

On the other hand by Pythagorean theorem from the right triangle we get:

$$L ^ {2} = h ^ {2} + 5 ^ {2}.$$

Substituting $h = L - 1$ we obtain

$$L ^ {2} = (L - 1) ^ {2} + 5 ^ {2},$$ $$L ^ {2} = L ^ {2} - 2 L + 1 + 2 5,$$ $$2 L = 2 6,$$ $$L = 1 3 \text {f e e t}.$$

Answer. $L = 13$ feet.