A car travels along a straight road, heading east for 1 h, then traveling for 30 min on another road that leads northeast. If the car has maintained a constant speed of 56 mi/h, how far is it from its starting position?
1.5 * 56 = ?
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To find out the distance of the car from its starting position, we need to break down the car's motion into two components: the eastward motion and the northeastward motion.
1. Eastward motion: The car travels east for 1 hour at a speed of 56 mi/h. Speed is defined as distance traveled per unit time. Therefore, the distance traveled eastward can be calculated by multiplying the speed by the time:
Distance eastward = Speed * Time = 56 mi/h * 1 h = 56 miles.
2. Northeastward motion: The car travels northeast for 30 minutes (which is equivalent to half an hour) at a speed of 56 mi/h. Since the car is moving northeast, it is essentially splitting its motion into two components: one to the north and one to the east. These components can be found using trigonometry.
The car is traveling at a constant speed of 56 mi/h, and the northeast direction can be broken down into two perpendicular components: one going east and one going north. Since the car is traveling at a constant speed, the distance traveled eastward and northward should be equal.
Using trigonometry, we can find the components of the speed:
Eastward component = Speed * cos(45°) = 56 mi/h * cos(45°) ≈ 39.6 mi/h.
Northward component = Speed * sin(45°) = 56 mi/h * sin(45°) ≈ 39.6 mi/h.
Since the car traveled for 30 minutes, the distance traveled northeastward can be found by multiplying the speed by the time:
Distance northeastward = Speed * Time = 56 mi/h * 0.5 h = 28 miles.
To calculate the total distance from the starting position, we need to find the resultant vector of the eastward and northeastward components. Since these components are perpendicular, we can use the Pythagorean theorem:
Total distance = √(Distance eastward^2 + Distance northeastward^2)
= √(56^2 + 28^2)
= √(3136 + 784)
= √(3920)
≈ 62.65 miles.
Therefore, the car is approximately 62.65 miles from its starting position.