To support a tree damaged in a storm, a 12-foot wire is secured from the ground to the tree at a point 10 feet off the ground. The tree meets the ground at a right angle.At approximately what angle does the wire meet the ground?

To solve this problem, we can use trigonometry. Specifically, we can use the tangent function, which is defined as the ratio of the opposite side to the adjacent side in a right triangle.

1. First, identify the sides of the triangle. The wire forms the hypotenuse of the triangle, the height of the tree forms the opposite side, and the distance from the base of the tree to where the wire is secured to the ground forms the adjacent side.

2. We know the length of the hypotenuse (12 feet) and the length of the opposite side (10 feet). We want to find the angle at the base of the tree, which we'll call θ.

3. The tangent of an angle is defined as the length of the opposite side divided by the length of the adjacent side. So, we have:

   tan(θ) = opposite / adjacent tan(θ) = 10 / adjacent

4. But we don't know the length of the adjacent side. However, we can use the Pythagorean theorem to find it. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. So, we have:

   (hypotenuse)² = (opposite)² + (adjacent)² (12)² = (10)² + (adjacent)² 144 = 100 + (adjacent)² 44 = (adjacent)² adjacent = √44 ≈ 6.63 feet

5. Now we can find the tangent of the angle:

   tan(θ) = 10 / 6.63 ≈ 1.51

6. To find the angle θ, we take the inverse tangent (also known as the arctangent) of 1.51. Using a calculator, we find:

   θ = arctan(1.51) ≈ 56.31 degrees

So, the wire meets the ground at an angle of approximately 56.31 degrees.