I have no idea how to solve this one problem:
Round to two decimal places.
Two observers 25 feet apart, sight to the top of the tree. If the angle of elevation to the top of the tree from the observer closest to the tree is 30 degrees and the angle of elevation from the furthest observer is 20 degrees, how tall is the tree?
I got 98.48 feet but it is wrong, what is the correct answer and how would I go about solving this? Thank you!
Suppose the nearer observer is at a distance x, and the tree has height h. Then, it is clear that
h/x = tan30°
h/(x+25) = tan20°
Now, eliminate x and we see that
h/tan30° = h/tan20° - 25
h(cot20°-cot30°) = 25
h = 25/(cot20°-cot30°)
h = 24.62 ft
To solve this problem, you can use trigonometry and the concept of similar triangles.
Let's denote the height of the tree as 'h'.
First, draw a diagram to visualize the situation. You have two observers, one closer to the tree and one further away, creating a triangle with the tree.
Since you are given the angles of elevation, you can conclude that the triangle formed by the two observers and the top of the tree is a right triangle. The observer closest to the tree creates a 30-degree angle of elevation, and the furthest observer creates a 20-degree angle of elevation.
Now, let's label the shorter side of the triangle as 'a' (represents the distance between the observer closest to the tree and the tree), and the longer side as 'b' (represents the distance between the furthest observer and the tree).
Next, we need to use trigonometry to find an equation relating the height 'h' of the tree with the given angles and distances.
For the observer closest to the tree, we can write the equation: tan(30°) = h/a. Rearranging this equation, we get a = h/tan(30°).
Similarly, for the furthest observer, we can write the equation: tan(20°) = h/b. Rearranging this equation, we get b = h/tan(20°).
Now, we need to use the given information that the two observers are 25 feet apart. Therefore, a + b = 25.
Substituting the expressions for a and b derived earlier into this equation, we get:
h/tan(30°) + h/tan(20°) = 25.
To solve this equation for 'h', follow these steps:
1. Convert the angles from degrees to radians by multiplying by π/180. The equation becomes:
h/(tan(30° * π/180)) + h/(tan(20° * π/180)) = 25.
2. Simplify the equation using a calculator:
h/(0.577) + h/(0.364) = 25.
3. Combine the fractions:
(1.732h + 2.747h)/(0.577 * 0.364) = 25.
4. Simplify further:
4.479h = 25 * 0.577 * 0.364.
5. Solve for 'h':
h = (25 * 0.577 * 0.364) / 4.479.
6. Use a calculator to evaluate the expression:
h ≈ 2.967 feet.
Rounding to two decimal places, the height of the tree is approximately 2.97 feet.
Therefore, the correct answer is 2.97 feet, not 98.48 feet.