Sure, let's solve this problem step by step.
1. Identify the Given Information:
- The distance from the observer (O) to the building (B) is [tex]\(900\)[/tex] feet.
- The angle of elevation from the observer's line of sight to the helicopter (H) is [tex]\(49^\circ\)[/tex].
2. Visualize and Label the Problem:
- Imagine a right-angled triangle where:
- The observer (O) is at one vertex.
- The base of the building (B) is 900 feet away from the observer (adjacent side of the triangle).
- The helicopter (H) is directly above the building, forming the right angle with the ground.
- The angle of elevation [tex]\(\angle O\)[/tex] from the observer's line of sight to the helicopter is [tex]\(49^\circ\)[/tex].
3. Use Trigonometry to Set Up the Equation:
- We need to find the height of the helicopter above the building (opposite side of the triangle, which we'll call [tex]\(h\)[/tex]).
- We can use the tangent function, which relates the opposite side to the adjacent side in a right-angled triangle:
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
where [tex]\(\theta\)[/tex] is the angle of elevation.
4. Substitute the Known Values into the Tangent Function:
- Here [tex]\(\theta = 49^\circ\)[/tex] and the adjacent side is 900 feet.
[tex]\[
\tan(49^\circ) = \frac{h}{900}
\][/tex]
5. Solve for [tex]\(h\)[/tex] (the height of the helicopter):
- Rearranging the equation to solve for [tex]\(h\)[/tex]:
[tex]\[
h = 900 \times \tan(49^\circ)
\][/tex]
- To find the tangent of [tex]\(49^\circ\)[/tex], we can use a calculator or a trigonometric table:
[tex]\[
\tan(49^\circ) \approx 1.15037
\][/tex]
6. Calculate the Height:
- Now, multiply the tangent of the angle by the distance to the building:
[tex]\[
h = 900 \times 1.15037 \approx 1035.33 \text{ feet}
\][/tex]
So, the height of the helicopter flying over the building is approximately 1035.33 feet.
