To determine who is farther from home, we'll break down the distances traveled by both you and your brother. You and your brother begin at home and together drive 25 miles due north to the airport. Subsequently, you each fly to your respective destinations. You travel 225 miles 15° toward the west, while your brother journeys 225 miles 25° toward the east. To compare the distances, we can employ trigonometry, decomposing the distances into their north-south and east-west components.
For you, the north-south component of your flight is 225 miles * sin(15°), which equates to 57.81 miles. The east-west component is 225 miles * cos(15°), resulting in 215.65 miles. For your brother, the north-south component of his flight is 225 miles * sin(25°), or 95.36 miles, and the east-west component is 225 miles * cos(25°), approximately 206.84 miles.
Now, we can calculate the total distances from home for both of you. For you, the total distance from home is the sum of the 25-mile drive to the airport and the distance covered during the flight. Thus, your total distance from home equals 25 miles + 215.65 miles, or 240.65 miles. For your brother, the total distance from home includes the 25-mile drive to the airport and the distance traveled during his flight. Thus, your brother's total distance from home is 25 miles + 206.84 miles, equating to 231.84 miles.
Therefore, you are farther from home with a total distance of 240.65 miles, compared to your brother's total distance of 231.84 miles.