The sum of the interior angles of a polygon is 1980^ exto. Find the number of sides the polygon has. 11. The diagram below is a sketch of a rice field (not drawn to scale), use it to generate a survey

The sum of the interior angles $S$ of a polygon with $n$ sides is given by the formula:

We are given $S = 1980^\circ$:

Divide both sides by $180^\circ$:

Add 2 to both sides:

The polygon has $\boxed{13 \text{ sides}}$.

11. The diagram below is a sketch of a rice field (not drawn to scale), use it to generate a surveyor's field book. (All measurements are in metres)

A surveyor's field book records measurements taken along a baseline. The readings are typically recorded from the bottom of the page upwards, representing movement along the baseline from the starting point (X) to the end point (Y).

The baseline is XY, and its total length is 150 m.

• From X (0m): Starting point.
• At 30m from X: Left offset to F is 20m. Right offset to A is 20m.
• At 60m from X: Left offset to E is 50m.
• At 100m from X: Right offset to B is 40m.
• At 120m from X: Left offset to D is 45m. Right offset to C is 40m.
• To Y (150m): End point of the baseline.

12. A boy walk directly from point Q towards the foot of a vertical flag post 200m away. After conveying a distance of 140m, he observes the angle of elevation of the top of the flag post is $75^\circ$.

Initial distance from Q to the foot of the flag post = 200 m.
Distance walked by the boy towards the flag post = 140 m.
New distance from the boy to the foot of the flag post = $200 \text{ m} - 140 \text{ m} = 60 \text{ m}$.

Let $h$ be the height of the flag post.
The distance from the boy's new position to the foot of the flag post is 60 m.
The angle of elevation to the top of the flag post is $75^\circ$.
This forms a right-angled triangle where:

• The height of the flag post ($h$) is the opposite side to the angle of elevation.
• The distance from the boy to the foot of the flag post (60 m) is the adjacent side to the angle of elevation.

The tangent of an angle in a right-angled triangle is the ratio of the opposite side to the adjacent side:

Using a calculator, $\tan(75^\circ) \approx 3.73205$

Rounding to two decimal places:

The height of the flag post is approximately $\boxed{223.92 \text{ m}}$.