Final answer:
Aaron's speed on the mountain road was 22 miles per hour, and his speed on the freeway was 66 miles per hour. This was determined by setting up an equation based on the time spent on each road segment and the fact that the freeway speed was three times the mountain road speed.
Step-by-step explanation:
Aaron's trip to his mountain cabin can be split into two segments: one on the freeway and the other on the mountain road. To find his speeds on both, we need to set up equations that represent the entire trip and the relationship between the two speeds.Aaron drove on the freeway from 9:15 until 10:45. That's a duration of 1 hour and 30 minutes or 1.5 hours. He then drove on the mountain road from 10:45 to 11:15, which is 30 minutes or 0.5 hours. If we let x represent Aaron's speed on the mountain road in miles per hour, his speed on the freeway would be 3x, since it's three times as much.We use the fact that Distance = Speed × Time to set up our equations. For the freeway part, the distance covered is 3x × 1.5 hours. For the mountain road, the distance is x × 0.5 hours. The total distance of the trip is 110 miles, so the sum of the distances on both segments should equal 110 miles.
Now we can set up the equation: 3x × 1.5 + x × 0.5 = 110.
When we solve for x, we get:
4.5x + 0.5x = 1105x = 110x = 22
Thus, Aaron's speed on the mountain road is 22 miles per hour, and his speed on the freeway is 66 miles per hour (since it is 3 times 22 mph).