a person standing 100ft. from the base of the tree looks up to the top of the tree with an angle of elevation of 52. assuming that the persons eyes are 5ft above ground the ground how tall is the tree?
The tree top is 100*tan(52) feet higher than the person's eyes.
To get the height h, above ground,
h=100tan(52)+5 feet
= 133 feet, approximately
Thanks
To find the height of the tree, we can use trigonometry.
Let's consider the given information:
- The person is standing 100ft away from the base of the tree.
- The person's eyes are 5ft above the ground.
- The person is looking up at an angle of elevation of 52 degrees.
We can set up a right triangle to represent the situation, with:
- The base of the triangle representing the distance from the person to the base of the tree (100ft).
- The height of the triangle representing the height of the tree (unknown).
- The hypotenuse of the triangle representing the distance from the person to the top of the tree (unknown).
Using trigonometry, specifically the tangent function (tan), we can determine the height of the tree.
Let's proceed step-by-step to calculate the height of the tree:
Step 1: The height of the right triangle can be calculated as:
Height of the triangle = Distance from the base of the triangle * tan(angle of elevation)
Height of the triangle = 100ft * tan(52 degrees)
Now, let's calculate the height of the triangle:
Height of the triangle = 100ft * tan(52 degrees)
Height of the triangle ≈ 128.77ft
Step 2: However, this height is measured from the person's eye level (5ft) above the ground. To get the actual height of the tree, we need to add the person's eye level height to the height of the triangle.
Actual height of the tree = Height of the triangle + Person's eye level height
Actual height of the tree = 128.77ft + 5ft
Finally, let's calculate the actual height of the tree:
Actual height of the tree ≈ 133.77ft
Therefore, the tree is approximately 133.77ft tall.
To find the height of the tree, we can use trigonometry and the concept of tangent. The tangent of an angle of elevation is defined as the length of the opposite side divided by the adjacent side.
In this case, the person's eyes can be considered as the point of reference, which is 5ft above the ground. The vertical distance from the person's eyes to the top of the tree is equal to the height of the tree. The horizontal distance from the person's eyes to the base of the tree is given as 100ft.
Now, let's solve the problem step by step:
Step 1: Identify the values given in the problem:
- Distance from the base of the tree to the person = 100ft
- Angle of elevation = 52 degrees
- Height of the person's eyes = 5ft
Step 2: Draw a diagram:
- Draw a triangle with the person's eyes, the top of the tree, and the base of the tree.
- Label the distance from the base of the tree to the person as 100ft, the angle of elevation as 52 degrees, and the height of the person's eyes as 5ft.
- Label the height of the tree as "h" (what we need to find).
Step 3: Identify the trigonometric ratio:
- In this case, we need to find the tangent of the angle of elevation because we are given the opposite (height of the tree) and the adjacent (distance from the base of the tree to the person).
Step 4: Write the equation:
- The tangent of the angle of elevation is equal to the ratio of the height of the tree to the distance from the base of the tree to the person.
- tan(52 degrees) = height of the tree / 100ft
Step 5: Solve the equation:
- Multiply both sides of the equation by 100ft to isolate the height of the tree.
- height of the tree = tan(52 degrees) * 100ft
Now we can calculate the height of the tree using a calculator or software that can compute trigonometric functions.
Using the equation we derived, the height of the tree is approximately 121.23ft.