To solve this problem, we need to break down the car's motion into two components: one for the eastward travel and another for the northeastward travel.

Step 1: Calculate the distance for each leg of the journey.

• Eastward Travel: The car travels east for 1 hour at a speed of 48 miles per hour. The distance covered during this part of the trip is: $\text{Distance east} = 48 \, \text{mi/h} \times 1 \, \text{hour} = 48 \, \text{miles}.$

• Northeastward Travel: The car then travels northeast for 30 minutes (which is 0.5 hours) at the same speed of 48 miles per hour. The distance covered during this part is: $\text{Distance northeast} = 48 \, \text{mi/h} \times 0.5 \, \text{hours} = 24 \, \text{miles}.$ Since the direction is northeast, the motion is at a 45° angle to the eastward direction. Therefore, the eastward and northward components of this distance are the same.

  The horizontal (eastward) and vertical (northward) components of the 24 miles are: $\text{East component} = 24 \times \cos(45^\circ) = 24 \times \frac{\sqrt{2}}{2} \approx 24 \times 0.7071 = 16.97 \, \text{miles}.$ $\text{North component} = 24 \times \sin(45^\circ) = 24 \times \frac{\sqrt{2}}{2} \approx 16.97 \, \text{miles}.$

Step 2: Find the total eastward displacement.

The total eastward displacement is the sum of the eastward distances from both parts of the trip: $\text{Total eastward distance} = 48 + 16.97 = 64.97 \, \text{miles}.$

Step 3: Find the total northward displacement.

The total northward displacement comes only from the northeastward portion of the trip: $\text{Total northward distance} = 16.97 \, \text{miles}.$

Step 4: Calculate the total distance from the starting position.

Now we can use the Pythagorean theorem to find the straight-line distance (the resultant displacement) from the starting position. This is the hypotenuse of a right triangle with legs 64.97 miles (eastward displacement) and 16.97 miles (northward displacement): $\text{Distance from starting position} = \sqrt{(64.97)^2 + (16.97)^2}$ $= \sqrt{4228.57 + 288.94}$ $= \sqrt{4517.51} \approx 67.3 \, \text{miles}.$

Final Answer:

The car is approximately 67.3 miles from its starting position.

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Would you like more details or have any further questions?

Here are some related questions:

1. How can the concept of vector components be used to solve similar problems in physics?
2. What would the result be if the car had traveled at a different speed, say 60 mi/h?
3. How does the Pythagorean theorem apply to real-world navigation problems?
4. How do you calculate displacements in problems involving circular motion?
5. What would be the effect if the car traveled for a longer time in the northeast direction?

Tip: Always break complex motion into horizontal and vertical components to simplify the calculations.