An aircraft is flying in a straight line away from a person standing on the ground. The initial angle of elevation changes from 70 degrees to 30 degrees in 1 minute. What is the cruising speed of the aircraft if its altitude is 10,000 m?
If x is the horizontal distance away (in km),
x/10 = cotθ
x = 10cotθ
x(70) = 3.63
x(30) = 17.32
so, the plane traveled 13.69km in 1 minute
speed is thus 13.69km/min
To find the cruising speed of the aircraft, we can use trigonometry and the concept of velocity. Let's break down the problem step by step:
Step 1: Understand the problem
We have an aircraft flying in a straight line away from a person standing on the ground. The initial angle of elevation changes from 70 degrees to 30 degrees in 1 minute. The altitude of the aircraft is given as 10,000 m. We need to calculate the cruising speed of the aircraft.
Step 2: Draw a diagram
Let's draw a diagram illustrating the situation. Consider a right-angled triangle with the person on the ground, the aircraft, and its altitude.
/|
/ |
/ | altitude
/ |
/____|
person | aircraft
on the |
ground |
angle 70° | angle 30°
Step 3: Identify relevant trigonometric ratios
We need to consider the trigonometric ratios involving the angle of elevation, the altitude, and the distance traveled by the aircraft.
The tangent of an angle is the ratio of the opposite side to the adjacent side. In this case, the opposite side is the altitude, the adjacent side is the distance traveled by the aircraft, and the angle is 30 degrees.
Recall that the tangent ratio is given by: tan(angle) = opposite / adjacent
Step 4: Solve the equation
Let's solve the equation for the tangent of 30 degrees:
tan(30°) = altitude / distance
Given that the altitude is 10,000 m, we have:
tan(30°) = 10,000 / distance
Next, calculate the tangent of 30 degrees using a scientific calculator:
tan(30°) ≈ 0.577
Substituting this value into the equation:
0.577 = 10,000 / distance
To isolate the distance, we rearrange the equation:
distance = 10,000 / 0.577
Simplifying:
distance ≈ 17319.96 m
Step 5: Calculate the time
The problem states that the plane changes altitude from 70 degrees to 30 degrees in 1 minute. This indicates a change in height, not distance. Therefore, we do not need to consider time in our calculation.
Step 6: Determine cruising speed
The cruising speed of the aircraft refers to its speed relative to the ground. Since we only have information about the aircraft's altitude, we cannot determine its cruising speed without additional data such as wind speed.
In conclusion, using the given information, we can determine the distance traveled by the aircraft but not its cruising speed.