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 32 mi/h, how far is it from its starting position? (Round your answer to one decimal place.)

To solve this problem, we need to determine the car's final position relative to its starting point after it travels in two different directions.

1. **Eastward Travel:**
   - The car travels east for 1 hour at 32 miles per hour.
   - Distance traveled east = speed × time = \( 32 \text{ mi/h} \times 1 \text{ h} = 32 \text{ miles} \).

2. **Northeastward Travel:**
   - The car then travels northeast (45 degrees from east) for 30 minutes at 32 miles per hour.
   - First, convert 30 minutes to hours: \( 30 \text{ min} = 0.5 \text{ h} \).
   - Distance traveled northeast = speed × time = \( 32 \text{ mi/h} \times 0.5 \text{ h} = 16 \text{ miles} \).

   Since the northeast direction is at a 45-degree angle, the distance traveled northeast can be broken down into its east and north components using trigonometric functions:
   - East component: \( 16 \cos(45^\circ) = 16 \times \frac{\sqrt{2}}{2} = 16 \times 0.7071 = 11.3 \text{ miles} \).
   - North component: \( 16 \sin(45^\circ) = 16 \times \frac{\sqrt{2}}{2} = 16 \times 0.7071 = 11.3 \text{ miles} \).

3. **Final Position:**
   - Total eastward distance = distance traveled east + east component of northeast travel = \( 32 \text{ miles} + 11.3 \text{ miles} = 43.3 \text{ miles} \).
   - Total northward distance = north component of northeast travel = \( 11.3 \text{ miles} \).

4. **Distance from Starting Position:**
   - The car's distance from the starting point can be found using the Pythagorean theorem:
   \[
   \text{Distance} = \sqrt{(\text{eastward distance})^2 + (\text{northward distance})^2}
   \]
   \[
   \text{Distance} = \sqrt{(43.3 \text{ miles})^2 + (11.3 \text{ miles})^2}
   \]
   \[
   \text{Distance} = \sqrt{1874.89 + 127.69}
   \]
   \[
   \text{Distance} = \sqrt{2002.58}
   \]
   \[
   \text{Distance} \approx 44.8 \text{ miles}
   \]

Thus, the car is approximately 44.8 miles from its starting position.

Would you like further details or have any questions?

Here are some related questions for further practice:

1. What would be the car's final position if it traveled 1 hour south instead of east initially?
2. How far would the car be from its starting position if it traveled at 40 mi/h instead of 32 mi/h?
3. If the car traveled north for the first hour instead of east, what would be its distance from the starting point?
4. What if the car traveled 45 minutes northeast and 45 minutes northwest instead?
5. How does changing the angle of the northeast travel affect the final distance from the starting point?
6. What would be the final position if the second leg of the journey was west instead of northeast?
7. How would the final distance be affected if the second leg of the journey was southeast?
8. What if the car had traveled northeast at 45 mi/h for 30 minutes?

**Tip:** When dealing with vector components, breaking down the movement into horizontal and vertical components can simplify calculations and provide a clearer understanding of the problem.

Learn how to calculate the distance of a car from its starting position after traveling east and northeast using trigonometric components and vector analysis. This problem involves applying the Pythagorean theorem to determine the final distance. Suitable for high school students.

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