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**Q1. A fire at a building B is reported on telephone to two fire stations F _{1} and F_{2}, 10 km apart from each other on a straight road.**

**F _{1} observes that the fire is at an angle of 60Â° to the road and F_{2} observes that it is at angle of 45Â° from it. The station F_{1} sends its team. **

**(a) Why the team of station F _{1} was sent? (b) How much distance the station F_{1} team will have to travel? (c) Which mathematical concept is involved in the above problem? (d) By sending its team, which value is depicted by the fire-station F_{1}?**

**Sol. **Let BL be the perpendicular from B on F_{1}F_{2}.

(a) âˆµ BL âŠ¥ F_{1}F_{2}

and âˆ LF_{1}B = 60Â°;

âˆ LF_{2}B = 45Â°

Ã¢ From right-angled Î”F_{1}LB and Î”F_{2}LB, we have:

â‡’

âˆ´ sinÎ¸ increases as Î¸ increases from 0Â° to 90Â°.

âˆ´ sin 45Â° < sin 60Â°

â‡’

âˆ´ Distance of B from F2 is more than that of from F1. That is why the station F1 sent its team to B.

Let F1B be x km

â‡’x cos 60Â° = F_{1}L ..........(1)

â‡’ BL = BF_{1} sin 60Â°

â‡’ F_{2}L = x sin 60Â° ..........(2)

........(3)

From (2) and (3), we get

F_{1}L + F_{2}L = 10

x cos 60Â° + x sin 60Â° = 10

â‡’ x [cos 60Â° + sin 60Â°] = 10

âˆµ Station F_{1} team travelled 7.32 km

(c) Trigonometry [Heights and Distances]

(d) Promptness.

**Q2. The angle of elevation of the top of a chimney from the foot of a tower is 60Â° and the angle of depression of the foot of the chimney from the top of the tower is 30Â°. If the height of the tower is 40m, find the height of the chimney. According to pollution control norms, the minimum height of a smoke emitting chimney should be 100m. State if the height of the above mentioned chimney meets the pollution norms. What value is discussed in this question?**

**Sol.** In the figure, the height of the tower (AB) = 40 m

Let the height of the chimney (CD) = h metre.

Thus, the height of the chimney is 120 m.

As, the minimum height of a chimney (according to the pollution control norms) should be 100 m

âˆ´ 120 m > 100 m

âˆ´ Thus, the above mentioned chimney meets the pollution norms.

Value : To keep pollution free environment.

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