While cruising at a steady speed of 400 km/h, you identify a storm cloud straight ahead 45 km away. To avoid turbulence, you start climbing at an angle of elevation of 15°. If you maintain this speed and direction for 6 min, how far will you be from the storm?
horizontal distance on original course, or line-of-sight distance?
If the latter, then draw a diagram. You have two sides of 45 and 40, and the angle between them is 15°. You want to find the third side.
just use the law of cosines to find x:
x^2 = 40^2 + 45^2 - 2*40*45*sin15°
To determine how far you will be from the storm, we need to calculate the horizontal distance covered during the climb.
First, we can find the vertical distance covered during the climb using trigonometry.
Vertical distance = Speed * Time * Sin(angle of elevation)
Given:
Speed = 400 km/h
Time = 6 minutes = 6/60 = 0.1 hours
Angle of elevation = 15°
Vertical distance = 400 km/h * 0.1 hours * sin(15°)
Vertical distance = 400 km/h * 0.1 hours * 0.259
Vertical distance = 10.36 km
Next, we can find the horizontal distance covered during the climb using trigonometry.
Horizontal distance = Speed * Time * Cos(angle of elevation)
Horizontal distance = 400 km/h * 0.1 hours * cos(15°)
Horizontal distance = 400 km/h * 0.1 hours * 0.966
Horizontal distance = 38.64 km
To find the remaining distance from the storm, we can subtract the horizontal distance from the initial distance.
Remaining distance = Initial distance - Horizontal distance
Given:
Initial distance = 45 km
Horizontal distance = 38.64 km
Remaining distance = 45 km - 38.64 km
Remaining distance = 6.36 km
Therefore, after 6 minutes of climbing at an angle of elevation of 15°, you will be approximately 6.36 km away from the storm.
To find out how far you will be from the storm cloud after 6 minutes, we need to break down the given information step by step.
Step 1: Convert the cruising speed from km/h to km/min.
Since the speed is given in km/h, we need to convert it to km/min because we are given the time in minutes. To convert km/h to km/min, we divide by 60 because there are 60 minutes in an hour.
400 km/h ÷ 60 = 6.67 km/min
So, the speed is approximately 6.67 km/min.
Step 2: Calculate the horizontal distance you traveled in 6 minutes.
Since you are traveling at a steady speed of 6.67 km/min, the horizontal distance you traveled can be determined by multiplying the speed by the time.
Horizontal distance = Speed × Time
Horizontal distance = 6.67 km/min × 6 min
Horizontal distance = 40 km
So, you traveled a horizontal distance of 40 km in 6 minutes.
Step 3: Calculate the vertical distance you climbed in 6 minutes.
To calculate the vertical distance, we need to use the trigonometric function tangent since the angle of elevation is given.
Vertical distance = Horizontal distance × tan(angle of elevation)
Note: Make sure the angle of elevation is in radians. If the angle is given in degrees, convert it to radians using the formula: radians = degrees × π/180.
The given angle of elevation is 15°. Converting it to radians:
15° × π/180 ≈ 0.262 radians
Now, we can calculate the vertical distance:
Vertical distance = 40 km × tan(0.262)
Vertical distance ≈ 10.91 km
So, you climbed a vertical distance of approximately 10.91 km in 6 minutes.
Step 4: Calculate the distance from the storm cloud.
To find the total distance from the storm cloud, we need to use the Pythagorean theorem, which states that the square of the hypotenuse (total distance) is equal to the sum of the squares of the other two sides (horizontal distance and vertical distance).
Total distance = √(Horizontal distance² + Vertical distance²)
Total distance = √(40² + 10.91²)
Total distance ≈ √(1600 + 119.1881)
Total distance ≈ √1719.1881
Total distance ≈ 41.47 km
So, after 6 minutes, you will be approximately 41.47 km away from the storm cloud.