Question

a plane is flying 2000 feet above sea level towards the mountain. The pilot observes the top of the mountain to be 18 degrees above the horizontal, then immediately flies the plane at an an angle of 20 degrees above horizontal. The airspeed of the plane is 100 mph. After 5 minutes, the plane is directly above the top if the mountain. How high is the plane above the top of the mountain (when it passes over)? What is the height of the mountain?

Answer 1

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Height of the plane above the top of the mountain = 16,575.08 feet

Height of the mountain = 22,995.61 feet


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To solve this problem, we can break it down into two parts: finding the height of the plane above the top of the mountain and calculating the height of the mountain itself.

Let's start with the height of the plane above the top of the mountain when it passes over.

1. Convert the airspeed of the plane from mph to feet per minute:

100 mph = (100 * 5280) feet per hour = 528,000 feet per hour

Therefore, the airspeed of the plane is 528,000 feet per hour / 60 minutes = 8,800 feet per minute.

2. Calculate the horizontal distance the plane travels in 5 minutes:
Speed = Distance / Time
Distance = Speed * Time
Distance = 8,800 feet per minute * 5 minutes = 44,000 feet

3. Calculate the vertical distance the plane travels while flying at an angle of 20 degrees above horizontal:
Vertical Distance = Horizontal Distance * tan(angle)
Vertical Distance = 44,000 feet * tan(20 degrees) ≈ 16,575.08 feet

Therefore, the height of the plane above the top of the mountain when it passes over is approximately 16,575.08 feet.

Now, let's calculate the height of the mountain:

4. Determine the horizontal distance from the observer (when the pilot observes the top of the mountain) to the mountain:
Horizontal Distance = Height / tan(angle)
Horizontal Distance = 2000 feet / tan(18 degrees) ≈ 6420.53 feet

5. Calculate the total height of the mountain:
Total Height = Height of Plane + Height from Observer to Mountain

Total Height = 16,575.08 feet + 6420.53 feet = 22,995.61 feet

Therefore, the height of the mountain is approximately 22,995.61 feet.

Answer 2

To find the height of the plane above the top of the mountain and the height of the mountain, we can use trigonometry and the distance formula. By solving the equations involved, we can determine the heights.

Step-by-step explanation:

To find the height of the plane above the top of the mountain, we can use trigonometry. Let's assume the height of the mountain is h and the horizontal distance from the plane to the mountain is x.

From the given information, we have:

• Angle of observation = 18 degrees
• Angle of the plane's path = 20 degrees
• Airspeed of the plane = 100 mph
• Time taken to reach above the top of the mountain = 5 minutes

Using trigonometry, we can write two equations:

Sin(18) = h/x

Sin(20) = (2000 + h)/x

By substituting h/x from the first equation into the second equation, we can solve for h. We can also use the formula Distance = Speed x Time to find the horizontal distance x, and then substitute it into one of the equations to find h. Therefore, we can calculate the height of the plane above the top of the mountain and the height of the mountain itself.