From a point on the ground 25 ft. from the foot of the tree, the angle of elevation of the top of the tree is 32 degrees. How do you find the height of the tree to the nearest foot?
The tree height is 25 tan32 = 15.6 feet
To the nearest foot, that would be 16.
I assume you are familiar with the tangent of an angle.
To find the height of the tree, you can use the tangent function, which relates the angle of elevation to the height and distance.
Here are the steps to find the height of the tree:
Step 1: Draw a diagram to visualize the problem. Label the distance from the point on the ground to the foot of the tree as the adjacent side (A), and the height of the tree as the opposite side (O).
/
/
/|
/ |
/ |O <- Height of the tree
/ |
/θ |
/____|
\ A /
Step 2: Use the tangent function: Tan(θ) = O / A, where Tan(θ) represents the tangent of the angle of elevation.
Step 3: Rearrange the equation to find the height of the tree (O): O = Tan(θ) * A.
Step 4: Plug in the known values:
- θ = 32 degrees (angle of elevation)
- A = 25 ft (distance from the point on the ground to the foot of the tree).
O = Tan(32) * 25
Step 5: Calculate the height using a calculator: O ≈ 14.411 ft.
Therefore, the height of the tree to the nearest foot is approximately 14 feet.
To find the height of the tree, you need to use trigonometry. The given angle of elevation and the known distance from the foot of the tree can be used to determine the height of the tree.
Here's how you can solve it:
Step 1: Draw a diagram representing the situation. Label the base of the tree as the foot, the top of the tree as the height, and the point on the ground where the person is standing as the observer.
Step 2: Identify the known and unknown values. In this case, the known values are the distance from the base of the tree to the observer (25 ft) and the angle of elevation (32 degrees). The unknown value is the height of the tree.
Step 3: Use the tangent function. The tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side. In this case, the opposite side is the height of the tree, and the adjacent side is the distance from the base of the tree to the observer.
The formula for the tangent function is:
tan(angle) = opposite side / adjacent side
In this problem, you can write:
tan(32 degrees) = height / 25 ft
Step 4: Solve for the height. To find the height, rearrange the equation and isolate the height:
height = 25 ft * tan(32 degrees)
Step 5: Use a calculator to evaluate the tangent of 32 degrees, and then multiply it by 25 ft to get the height of the tree. Round the answer to the nearest foot.
Using a calculator, tan(32 degrees) is approximately 0.6249. Multiplying this by 25 ft gives you:
height = 25 ft * 0.6249 ≈ 15.62 ft
Therefore, the height of the tree is approximately 15.62 feet when rounded to the nearest foot.
It'll help you sove this if you draw a picture.(Not the greatest below but you get the idea)
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. |x
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._____________|
25ft
The 32 degrees should go in the bottom left corner of the triangle. To solve for x, you would use tangent. The problem would be set up as tan(32)=x/25
Then, mulitply both sides by 25 to get x by itself so you have:
25tan(32)=x
When you plug this into a calculator, you get 16.525, so the tree is about 17ft tall