a plane is flying horizontally at a height of 10000m above a level stretch of ground at a constant speed of 240 km/h. A man on the ground observes the angle of elevation of the plane (which is flying on a course directly towards him) to be 35 degrees. how far (to the nearest km) in a straight line is the man from the plane, 1 minute after he measured the angle of elevation?
let the horizontal distance, along the ground, be x km
so 10 km/x km = tanØ
when Ø = 35°, 10/x = tan35
x = 14.2815 km
that distance x is decreasing at 240km/h
= 240km/60 m = 4 km/min
so 1 minute later, x = 14.2815 - 4 = 10.2815 km
hypotenuse distance^2 = 10^2 + 10.2815^2
hypotenuse distance = 14.343 km or 14 km to the nearest km
To solve this problem, we can use trigonometry. We'll use the tangent function to find the distance between the man and the plane.
Let's break down the problem into steps:
1. Convert the speed of the plane from kilometers per hour to meters per second. We know that 1 kilometer is equal to 1000 meters, and 1 hour is equal to 3600 seconds. So, the speed of the plane in meters per second is 240 km/h × (1000 m/1 km) × (1 h/3600 s).
Speed of the plane = (240 × 1000) / 3600 = 66.67 m/s (rounded to two decimal places)
2. Convert the time from minutes to seconds. Since we need to find the distance 1 minute after the observation, we convert 1 minute to seconds. Since 1 minute is equal to 60 seconds, the time is 60 seconds.
3. Now, let's find the horizontal distance the plane travels during that 1 minute. The horizontal distance is given by the formula: Distance = Speed × Time.
Distance = 66.67 m/s × 60 s = 4000.20 m (rounded to two decimal places)
4. Next, we'll find the vertical distance between the man and the plane. The vertical distance is equal to the height of the plane, which is 10,000 m.
Vertical distance = 10,000 m
5. Finally, we can use trigonometry to find the straight-line distance between the man and the plane. We'll use the tangent function since we have the opposite and adjacent sides of the right triangle formed by the man, the plane, and the ground.
The tangent of the angle of elevation is given by the formula: tan(angle) = Opposite / Adjacent.
In our case, the angle of elevation is 35 degrees, and the opposite side is the vertical distance (10,000 m). We need to find the adjacent side, which is the straight-line distance between the man and the plane.
tan(35 degrees) = 10,000 m / Adjacent
Rearranging the formula, we get:
Adjacent = 10,000 m / tan(35 degrees)
Now we can calculate the value of the adjacent side using a calculator or trigonometric table.
Approximate value of tan(35 degrees) = 0.7002 (rounded to four decimal places)
Adjacent = 10,000 m / 0.7002 = 14,280.77 m (rounded to two decimal places)
So, the man is approximately 14,280.77 meters away from the plane.
To get the answer in kilometers, we divide the distance by 1000:
14,280.77 m / 1000 = 14.28 km (rounded to two decimal places)
Therefore, the man is approximately 14.28 kilometers away from the plane 1 minute after measuring the angle of elevation.