To help a sapling (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 to the nearest tenth of a foot.
h = 2+10sin58°
To find the height of the tree, we can use trigonometry and the concept of similar triangles. Let's break down the problem step by step:
1. Draw a diagram: Draw a vertical line to represent the tree, and mark its height as 'h'. Label the wires as W1, W2, and W3, and their respective angles with the ground as θ1, θ2, and θ3.
2. Use trigonometry: In a right triangle, the sine of an angle is equal to the ratio of the opposite side length to the hypotenuse length. Since we have the lengths of the wires and the angles with the ground, we can use the sine function to relate the height of the tree to the length of the wires and their angles.
3. Using the sine function: For each wire, the opposite side length is the height of the tree (h), and the hypotenuse length is the length of the wire (10 feet). So we can write the following equations:
sin(θ1) = h / 10
sin(θ2) = h / 10
sin(θ3) = h / 10
4. Solve the equations: Rearrange each equation to solve for h:
h = 10 * sin(θ1)
h = 10 * sin(θ2)
h = 10 * sin(θ3)
5. Calculate the height: Substitute the given values into the equations and solve for h:
h = 10 * sin(58 degrees)
h = 10 * sin(58 degrees)
h = 10 * sin(58 degrees)
Using a calculator, we find that sin(58 degrees) ≈ 0.8480.
h ≈ 10 * 0.8480
h ≈ 8.48 feet
6. Round the height: Finally, we round the height to the nearest tenth, which gives us the height of the tree as approximately 8.5 feet.
Therefore, the height of the tree is approximately 8.5 feet.