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A radio survey is taken on a plot of land. Points BNC equal distance in kilometres from O. Point B is on a bearing of 315° and point C is on a bearing of 253°. The area of angle BOC is 148 km². What is the distance? OB correct to one decimal place?

To solve for the distance \( OB \) correct to one decimal place, we can use the formula for the area of a triangle with sides \( a \) and \( b \) and the angle \( C \) between them. The formula is:

\[
\text{Area} = \frac{1}{2}ab \sin C
\]

Given:
- Area of \(\triangle BOC = 148 \text{ km}^2\)
- Bearing of \( B \) from \( O \): \( 315^\circ \)
- Bearing of \( C \) from \( O \): \( 253^\circ \)
- Angle \( BOC \) is \( \theta = 315^\circ - 253^\circ = 62^\circ \)

Since \( B \) and \( C \) are equidistant from \( O \), let the distance \( OB = OC = r \). Then:

\[
\text{Area} = \frac{1}{2} r^2 \sin 62^\circ
\]

Solving for \( r \):

\[
148 = \frac{1}{2} r^2 \sin 62^\circ
\]

\[
r^2 = \frac{2 \times 148}{\sin 62^\circ}
\]

\[
r = \sqrt{\frac{296}{\sin 62^\circ}}
\]

Let’s compute the value of \( r \).The distance \( OB \) (and \( OC \), since they are equidistant from \( O \)) is \( 18.3 \) km, correct to one decimal place.

Would you like further details or have any questions?

### Additional Questions:
1. How can you calculate the distance \( BC \) given \( OB \) and \( OC \)?
2. What is the significance of the bearing in navigation and surveying?
3. How does the angle between two bearings affect the area of the triangle formed?
4. What methods can be used to verify the calculated distance \( OB \)?
5. Can the formula used here be applied to any triangle? If not, why?
6. How would the problem change if \( B \) and \( C \) were not equidistant from \( O \)?
7. What is the role of trigonometric functions in solving triangle-related problems?
8. How would you find the coordinates of point \( B \) given the distance \( OB \) and the bearing?

### Tip:
Always ensure that angles are measured correctly and consistently, especially when switching between different units (degrees and radians) or different systems (bearings and standard angles).

Learn how to calculate the distance OB to one decimal place using bearings and the area of triangle BOC. This problem involves applying trigonometric principles and the area formula of a triangle. Suitable for high school students in Grades 10-12.

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