A car travels on a bearing of 35° for 25 miles, then turns 90° to the right and travels for a further 35 miles before stopping.

Calculate the distance of the car from its starting point. Give your answer in miles, to 2 decimal places. and Calculate the bearing from the start point of the journey to the end point. Give your answer to the nearest degree.

We will solve this problem by applying basic trigonometry and vector calculations. Here's the breakdown:

### Part 1: Distance from the starting point

The car follows two movements:
1. **First leg**: 25 miles at a bearing of 35°.
2. **Second leg**: Turns 90° to the right and travels 35 miles.

We will treat this as a triangle and use the law of cosines to find the final distance from the starting point.

#### Step 1: Resolve the first leg into components
We can break the first leg into two components (north-south and east-west):
- North-South: \( 25 \times \cos(35^\circ) \)
- East-West: \( 25 \times \sin(35^\circ) \)

#### Step 2: Second leg
Since the car turns 90° to the right, the second leg is perpendicular to the first. This creates a right angle at the turn. The second leg is entirely horizontal (east-west), adding to the east-west displacement:
- North-South displacement: \( 25 \times \cos(35^\circ) \) (unchanged)
- East-West displacement: \( 25 \times \sin(35^\circ) + 35 \)

#### Step 3: Final distance using the Pythagorean theorem
We now calculate the total displacement using the Pythagorean theorem:
\[
\text{Total distance} = \sqrt{ \left( 25 \times \cos(35^\circ) \right)^2 + \left( 25 \times \sin(35^\circ) + 35 \right)^2 }
\]

Let's calculate that.The total distance of the car from its starting point is approximately **53.42 miles**.

### Part 2: Bearing from the start point to the end point

To calculate the bearing from the starting point to the end point, we use the formula for the bearing angle:
\[
\text{Bearing} = \arctan\left(\frac{\text{East-West displacement}}{\text{North-South displacement}}\right)
\]
This will give us the angle relative to the north-south line, which we will adjust based on the quadrant.

Let's calculate the bearing now.The bearing from the starting point to the end point is approximately **67°**.

### Summary:
- The distance of the car from the starting point is **53.42 miles**.
- The bearing from the starting point to the end point is **67°** (to the nearest degree).

Do you need further details or clarifications on the process?

Here are 5 related questions you might find helpful:
1. How would the result change if the second leg was longer or shorter?
2. How does the law of cosines apply when solving non-right triangles in general?
3. What if the bearing turned 45° instead of 90°? How would the calculation change?
4. Could this problem be solved graphically using a coordinate system?
5. How does the sine rule help in solving similar navigation problems?

**Tip:** Always double-check the units of your angles when switching between degrees and radians in trigonometry calculations.

A car travels on a bearing of 35° for 25 miles, then turns 90° to the right and travels for a further 35 miles before stopping. Calculate the distance of the car from its starting point. Give your answer in miles, to 2 decimal places. Also, calculate the bearing from the start point of the journey to the end point. Give your answer to the nearest degree.

Solve a two-leg car journey problem involving trigonometry to find the total distance (53.42 miles) and the bearing (67°) from the start point. Learn how to apply vector calculations, displacement components, and the Pythagorean theorem. Suitable for students in Grades 10-12.

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