To help a young tree grow straight, a gardener attaches three guy wires to it and the ground. She places the wires two feet below the top of the tree. If the wires are ten feet long and each makes an angle of 58 degrees with the ground. Find the height of the tree.
Consider one guy wire. Let x be the height at which the guy wire is attached to the tree.
sin58 = x/10
x = 10sin58 ft
Tree height = 10sin58 + 2 = ____ ft
To find the height of the tree, we can use trigonometry and the given information about the guy wires. Here's how you can solve it step by step:
1. Draw a diagram: Start by drawing a diagram that represents the situation described in the problem. Draw a straight vertical line to represent the height of the tree. Label it as the tree's height (h). Mark a point on the line two feet below the top to represent where the guy wires are attached.
2. Label the given information: Label the length of each guy wire as 10 feet and the angle it makes with respect to the ground as 58 degrees. Mark these angles on the diagram accordingly.
3. Use trigonometry: Trigonometry relates angles and sides of triangles. In this case, we can use the tangent function (tan) to relate the angle and the height of the tree.
The tangent of an angle in a right triangle is equal to the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.
4. Apply the tangent function: In the diagram, the guy wire is the side opposite to the angle, and the height of the tree is the side adjacent to the angle. So, we can use the tangent function to set up an equation:
tan(58 degrees) = height of tree / 10 feet
5. Solve for the height of the tree: To find the height of the tree, we can rearrange the equation:
height of tree = 10 feet * tan(58 degrees)
Using a calculator, calculate the value of tan(58 degrees) and multiply it by 10 feet to find the height of the tree.
height of tree ≈ 18.332 feet
Therefore, the height of the tree is approximately 18.332 feet.