A surveyor measuring the tallest tree in a park is 100ft from the tree. His angle-measuring device is 5 ft above the ground. The angle of elevation to the top of the tree is 48°. How tall is the tree?
Note that we can form a right triangle here, with base equal to 100 ft and angle between hypotenuse and base equal to 48°.
Let h = height of triangle, which is also equal to total height of statue minus the 5 ft elevation of the device. Thus,
tan 48° = h / 100
h = 100 * tan 48°
h = 111.1 ft
Therefore the height of tree is equal to h + 5 = 116.1 ft.
Hope this helps~ :3
Well, I must say, this surveyor really knows how to have a tree-mendous time! Let's calculate the height of the tree, shall we?
Now, we have a triangle formed by the surveyor, the top of the tree, and the base of the tree. Since we know the distance from the surveyor to the tree is 100ft and the angle of elevation is 48°, we can use some trigonometry to solve this arboreal mystery.
First, we need to find the distance from the surveyor's eye level to the top of the tree. This can be calculated using the tangent function: tangent(48°) = height of the tree / distance to the tree.
Using this formula, we get: tangent(48°) = height of the tree / 100ft.
Now, let's drop the punchline. After plugging in the values and solving for the height of the tree, we get approximately 105.99ft. So, according to my calculations, this tree is reaching for the skies at about 106 feet!
Remember, when it comes to measuring trees, always be prepared to branch out with some trigonometry!
To solve this problem, we can use the tangent function, which relates the angle of elevation to the height and the distance.
Step 1: Identify the known values:
- Distance from the tree to the surveyor: 100 ft.
- Height of the angle-measuring device: 5 ft.
- Angle of elevation: 48°.
Step 2: Draw a diagram:
Draw a right triangle with the following labels:
- The horizontal leg: represents the distance from the tree to the surveyor (100 ft).
- The vertical leg: represents the height of the tree (h).
- The hypotenuse: represents the distance from the angle-measuring device to the top of the tree (h + 5 ft).
Step 3: Apply the tangent function:
The tangent of the angle of elevation is equal to the opposite side (height of the tree) divided by the adjacent side (distance from the surveyor to the tree).
tangent(48°) = h / 100 ft
Step 4: Solve for h:
To isolate h, we can rearrange the equation:
h = tangent(48°) * 100 ft
Step 5: Calculate the height of the tree:
Using a calculator or the tangent table, find the tangent of 48°:
tangent(48°) ≈ 1.1106
Now substitute this value into the equation:
h ≈ 1.1106 * 100 ft
h ≈ 111.06 ft
Therefore, the height of the tree is approximately 111.06 feet.
To find the height of the tree, we can use trigonometry. Let's break down the problem and explain how to find the answer step by step:
1. Draw a diagram: It is always helpful to visualize the problem. Draw a triangle representing the situation. Label the distance between the surveyor and the tree as the base, the height of the tree as the opposite side, and the angle of elevation as the angle between the base and the opposite side.
2. Identify the known values: From the problem, we know the distance from the surveyor to the tree is 100 ft and the height of the angle-measuring device is 5 ft.
3. Determine the trigonometric function to use: In this case, we want to find the length of the opposite side (height of the tree) given the length of the adjacent side (distance to the tree) and the measure of the angle of elevation. This calls for using the tangent function.
4. Set up the equation: The tangent function is defined as the ratio of the opposite side to the adjacent side. Therefore, we have the equation:
tan(48°) = (height of the tree) / 100 ft
5. Solve for the height of the tree: Rearrange the equation to solve for the height of the tree:
(height of the tree) = tan(48°) * 100 ft
6. Calculate the height of the tree: Use a calculator to find the tangent of 48° and then multiply that value by 100 ft to get the height of the tree.
By following these steps, you can find the height of the tree.