A helicopter flies horizontally at 100 km/h. An observer notices that it takes 20 seconds for the helicopter to fly from directly overhead to being at an angle of elevation of 60 d... A helicopter flies horizontally at 100 km/h. An observer notices that it takes 20 seconds for the helicopter to fly from directly overhead to being at an angle of elevation of 60 degrees. Find the height of the helicopter above the ground.
Understand the Problem
The question asks us to calculate the height of the helicopter above the ground given its horizontal speed, the time it takes for the observer to see it at a 60-degree angle of elevation, using trigonometric relationships.
Answer
$\frac{5000\sqrt{3}}{9}$ m
Answer for screen readers
The height of the helicopter above the ground is $\frac{5000\sqrt{3}}{9}$ meters.
Steps to Solve
- Convert the speed from km/h to m/s
To be consistent with the time given in seconds, we need to convert the helicopter's speed from km/h to m/s
$100 \frac{km}{h} = 100 \cdot \frac{1000 m}{3600 s} = \frac{100000}{3600} \frac{m}{s} = \frac{1000}{36} \frac{m}{s} = \frac{250}{9} \frac{m}{s}$
- Calculate the horizontal distance covered in 20 seconds
Using the formula distance = speed $\times$ time, we can find the horizontal distance the helicopter covers in 20 seconds.
Distance $= \frac{250}{9} \frac{m}{s} \times 20 s = \frac{5000}{9} m$
- Use the tangent function to relate the height and the horizontal distance
Let $h$ be the height of the helicopter above the ground. The angle of elevation is $60^\circ$. The tangent of the angle of elevation relates the height ($h$) to the horizontal distance ($d$) from the observer to the point directly below the helicopter.
$\tan(60^\circ) = \frac{h}{d}$
Where $d$ represents our horizontal distance from step 2, $d = \frac{5000}{9} m$
$h = d \tan(60^\circ)$
Since $\tan(60^\circ) = \sqrt{3}$:
$h = \frac{5000}{9} \sqrt{3} m$
- Approximate the value (if needed)
$h = \frac{5000\sqrt{3}}{9} m \approx 962.25 m$
The height of the helicopter above the ground is $\frac{5000\sqrt{3}}{9}$ meters.
More Information
The height of the helicopter is approximately 962.25 meters. The problem combined concepts of speed, time, distance, and trigonometry to find the solution.
Tips
A common mistake would be not converting the speed from km/h to m/s, which would result in an incorrect distance calculation. Also, students might confuse the trigonometric ratios (sine, cosine, tangent) and use the wrong one, or incorrectly set up the ratio. For example, using $cos(60^\circ)$ or $sin(60^\circ)$ instead of $tan(60^\circ)$, or inverting the ratio. Another mistake could be misinterpreting the problem setup and not understanding which distance is opposite and adjacent to the given angle.
AI-generated content may contain errors. Please verify critical information
