(b) A person at the top of a lighthouse, $T B$ , sees two ships, $S_{1}$ and $S_{2}$ , approaching the coast as illustrated in the diagram below. The angles of depression are $12^{\circ}$ and $20^{\circ}$ respectively. The ships are $110 \mathrm{~m}$ apart. $\newline$ (i) Complete the diagram below by inserting the angles of depression and the distance between the ships. $\newline$ ( $1$ mark) $\newline$ (ii) Determine, to the nearest metre, $\newline$ a) the distance, $T S_{2}$ , between the top of the lighthouse and Ship $2$

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Q. (b) A person at the top of a lighthouse,

T B

, sees two ships,

S_{1}

and

S_{2}

, approaching the coast as illustrated in the diagram below. The angles of depression are

12^{\circ}

and

20^{\circ}

respectively. The ships are

110 \mathrm{~m}

apart.

\newline

(i) Complete the diagram below by inserting the angles of depression and the distance between the ships.

\newline

(

1

mark)

\newline

(ii) Determine, to the nearest metre,

\newline

a) the distance,

T S_{2}

, between the top of the lighthouse and Ship

2

Label angles and distance: Label the diagram with the angles of depression $12^\circ$ and $20^\circ$ , and the distance between the ships as $110\,\text{m}$ .
Find angle of elevation: Use the angle of depression to find the angle of elevation from Ship $2$ to the top of the lighthouse, which is also $20^\circ$ (angles of elevation and depression are equal).
Use tangent function: Let $TB$ be the height of the lighthouse, and $TS_2$ be the distance from the top of the lighthouse to Ship $2$ . Use the tangent function: $\tan(20°) = \frac{TB}{TS_2}$ .
Find height of lighthouse: We don't have TB, but we can find it using Ship $1$ . Let's call the distance from the top of the lighthouse to Ship $1$ as $TS_1$ . Using $\tan(12°) = \frac{TB}{TS_1}$ .
Calculate distance $TS_1$ : Rearrange the equation to find $TB$ : $TB = TS_1 \cdot \tan(12^\circ)$ .
Calculate distance $TS_1$ : Rearrange the equation to find $TB$ : $TB = TS_1 \cdot \tan(12°)$ .We need to find $TS_1$ first. Since the ships are $110m$ apart, and we have two angles of elevation, we can use the angle difference to find $TS_1$ . Let's call the angle between the ships as seen from the lighthouse $\theta$ . Then $\theta = 20° - 12° = 8°$ .
Calculate distance $TS_1$ : Rearrange the equation to find $TB$ : $TB = TS_1 \cdot \tan(12°)$ .We need to find $TS_1$ first. Since the ships are $110m$ apart, and we have two angles of elevation, we can use the angle difference to find $TS_1$ . Let's call the angle between the ships as seen from the lighthouse $\theta$ . Then $\theta = 20° - 12° = 8°$ .Now, we can find $TS_1$ using the Law of Sines in the triangle formed by the lighthouse, Ship $1$ , and Ship $2$ . $\frac{\sin(\theta)}{110} = \frac{\sin(12°)}{TS_1}$ .
Calculate distance $TS_1$ : Rearrange the equation to find $TB$ : $TB = TS_1 \cdot \tan(12^\circ)$ .We need to find $TS_1$ first. Since the ships are $110\,\text{m}$ apart, and we have two angles of elevation, we can use the angle difference to find $TS_1$ . Let's call the angle between the ships as seen from the lighthouse $\theta$ . Then $\theta = 20^\circ - 12^\circ = 8^\circ$ .Now, we can find $TS_1$ using the Law of Sines in the triangle formed by the lighthouse, Ship $1$ , and Ship $2$ . $\frac{\sin(\theta)}{110} = \frac{\sin(12^\circ)}{TS_1}$ .Rearrange to find $TS_1$ : $TB$ $1$ .
Calculate distance $TS_1$ : Rearrange the equation to find $TB$ : $TB = TS_1 \cdot \tan(12°)$ .We need to find $TS_1$ first. Since the ships are $110m$ apart, and we have two angles of elevation, we can use the angle difference to find $TS_1$ . Let's call the angle between the ships as seen from the lighthouse $\theta$ . Then $\theta = 20° - 12° = 8°$ .Now, we can find $TS_1$ using the Law of Sines in the triangle formed by the lighthouse, Ship $1$ , and Ship $2$ . $\frac{\sin(\theta)}{110} = \frac{\sin(12°)}{TS_1}$ .Rearrange to find $TS_1$ : $TB$ $1$ .Calculate $TS_1$ : $TB$ $3$ .