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 60° to the road and F_{2} observes that it is at angle of 45° 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 firestation F_{1}?
Sol. Let BL be the perpendicular from B on F_{1}F_{2}.
(a) ∵ BL ⊥ F_{1}F_{2}
and ∠ LF_{1}B = 60°;
∠ LF_{2}B = 45°
â From rightangled ΔF_{1}LB and ΔF_{2}LB, we have:
⇒
∴ sinθ increases as θ increases from 0° to 90°.
∴ sin 45° < sin 60°
⇒
∴ 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
⇒x cos 60° = F_{1}L ..........(1)
⇒ BL = BF_{1} sin 60°
⇒ F_{2}L = x sin 60° ..........(2)
........(3)
From (2) and (3), we get
F_{1}L + F_{2}L = 10
x cos 60° + x sin 60° = 10
⇒ x [cos 60° + sin 60°] = 10
∵ 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 60° and the angle of depression of the foot of the chimney from the top of the tower is 30°. 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
∴ 120 m > 100 m
∴ Thus, the above mentioned chimney meets the pollution norms.
Value : To keep pollution free environment.
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