A plane leaves an airport and flies at 36 mph on a course of 103.25° for 1 hour and 33 minutes. Then it changes course to 236.75° and flies for 48 minutes. How far is it from the airport? If it wished to return, what would be its course?
Draw a diagram. Use the law of cosines, since you have two sides and the angle between them, and want to find the 3rd side.
To get you started, the angle between the two sides is 33.25+13.25=47.5°
actually, that's (270-236.25)+(103.25-90).
To get the side lengths, remember that distance=speed*time
To find how far the plane is from the airport, we can use the concept of vector addition.
1. Convert the time values to a common unit. Given that the first duration is 1 hour and 33 minutes, we convert it to 1.55 hours (1 hour and 33 minutes is equal to 1 + 33/60 = 1.55 hours). Similarly, the second duration of 48 minutes can be converted to 0.8 hours.
2. Calculate the distance covered in the first part of the journey. This can be done by multiplying the speed of 36 mph by the duration of 1.55 hours. So, the distance covered in the first part is 36 mph * 1.55 hours = 55.8 miles.
3. Determine the components of the first leg of the journey. The course of 103.25° can be divided into horizontal and vertical components. The horizontal component is given by cos(103.25°) * 55.8 miles and the vertical component is given by sin(103.25°) * 55.8 miles.
4. Calculate the distance covered in the second part of the journey. Similar to the first leg, the speed is 36 mph and the duration is 0.8 hours. So, the distance covered in the second part is 36 mph * 0.8 hours = 28.8 miles.
5. Determine the components of the second leg of the journey. The course of 236.75° can be divided into horizontal and vertical components. The horizontal component is given by cos(236.75°) * 28.8 miles and the vertical component is given by sin(236.75°) * 28.8 miles.
6. Add the horizontal and vertical components of both legs to find the final position of the plane from the starting point. Add the horizontal components and vertical components separately to get the total horizontal distance and vertical distance. Then, use the Pythagorean theorem to calculate the total distance from the airport.
7. To calculate the angle for the return course, subtract the original course from 180°. This assumes the original course was measured from the north, and 180° represents the opposite direction.
Following these steps, you can determine the distance of the plane from the airport and the course for its return journey.