On a map, point C is 4.3 km due east of point B, whereas point B is 2.7 km on a true bearing of 143° from point A. Give your answer to two decimal places for the following.
a Find how far east point B is from A.

To find how far east point B is from point A, we can use trigonometry. Since point B is located on a true bearing of 143° from point A, we can break down the displacement into its eastward and southward components using the sine and cosine functions.

Given:
- The distance from point A to point B is 2.7 km.
- The bearing from point A to point B is 143°.

To find the eastward distance (which we'll call \(x\)) from point A to point B, we use the cosine of the angle. However, since 143° is measured clockwise from the north, the eastward component will be found using the angle with respect to the south direction (180° - 143° = 37°).

The eastward component is given by:
\[ x = 2.7 \times \sin(37°) \]

Calculating this:
\[
x = 2.7 \times \sin(37°)
\]
\[
x \approx 2.7 \times 0.6018 \quad (\text{using a calculator})
\]
\[
x \approx 1.62 \text{ km}
\]

So, point B is approximately 1.62 km east of point A.

Do you want any further details or have any questions?

Here are 8 related questions to expand the information:
1. What is the total distance from point A to point C via point B?
2. How far south is point B from point A?
3. What is the true bearing from point A to point C?
4. What is the angle between the line AB and the eastward direction?
5. If point D is 3 km north of point C, what is the distance from point A to point D?
6. How would you represent these points and bearings on a coordinate plane?
7. What is the midpoint between point A and point B?
8. How far is point C from point A in a straight line?

**Tip:** When working with bearings and trigonometric calculations, always double-check your angle conversions and ensure your calculator is set to the correct mode (degrees or radians) based on the given information.

Learn how to calculate the eastward distance from point A to point B using trigonometry and bearings. This problem involves finding the eastward component using the sine function with a given bearing of 143° and a distance of 2.7 km. Suitable for students in Grades 9-12.

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