# Examples I

The angle of elevation of the top of a tower, at a distance 100 m from its foot on a horizontal plane, is found to be 60Â°. Find the height of the tower.

Let AB (= h) be the height of the tower.

In the triangle tan 60Â° =Â

â‡’Â Â Â Â Â Â Â =Â

â‡’Â Â Â Â Â Â Â Â h = 100Â = 173.2 m.

Hence, the height of the tower is 173.2 m.

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A ladder leaning against a wall makes an angle of 20Â° with the ground. The foot of the ladder is 3 m away from the wall. Determine the length of the ladder.

Â

Let AC (= l) be the length of the ladder in m. In the triangle ABC,Â *Â *= cos 20Â°Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â

Hence, the length of the ladder is 3.192 m.

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The shadow of a tower standing on a flat ground is found to beÂ Â longer, when the Sun's altitude is 30Â°, than when it was 60Â°. Find the height of the tower.

Let CD = h m be the height of the tower, let x be the length of the shadow when Sun's altitude = 60Â°.

Length of the shadow =x + 45Â when Sun's altitude = 30Â°

In theÂ Î”BCD,

â‡’Â Â =Â Â â‡’Â Â Â x =Â Â Â Â ....(1)

In theÂ Î”Â ACD,

Â = tan 30Â°

Substituting for x from (1)

â‡’Â Â Â

â‡’Â Â

â‡’Â Â Â 3h = h + 135

â‡’Â Â h = 67.5 m.

Hence, the height of the tower is 67.5 m.

Â

Â

Two men standing on the opposite sides of a flag staff measure the angles of the top of the flag-staff as 30Â° and 60Â°. If the height of the flag staff is 20 m, find the distance between the men.

Let BC be the flag staff and A and D be the positions of the two men on either sides of the flag staff.

BC = 20 m.Â

In theÂ Î”Â ABC, tan 30Â°=Â

â‡’Â Â Â Â Â Â Â Â Â Â

â‡’Â Â Â Â Â Â Â Â Â Â Â AB = 20Â m

In theÂ Î”Â DBC, tan 60Â°=Â

â‡’Â Â Â Â Â Â Â Â Â Â Â Â =Â

â‡’Â Â Â Â Â Â Â Â Â Â Â Â Â BD =Â

NowÂ AD = AB + BD

Â Â Â Â Â Â Â Â Â Â Â =Â 20

Â Â Â Â Â Â Â Â Â Â Â =Â = 46.19

â‡’Â The distance between the men is 46.19 m.

Â

A flagstaff stands on a top of a 10 m tower. From a point on the ground the angle of elevation of the top of the flagstaff is 60Â° and that of the tower is 45Â°. Find the distance of the point of observation (on the ground) from the top of the flagstaff. Also find the height of the flagstaff.

Let BC be the tower and CD the flagstaffÂ on top of the tower. From the point of observation A the flagstaff is x m away.

In theÂ Î”Â ABC,Â Â = tan 45Â°.

â‡’Â Â Â Â Â Â Â Â Â Â Â Â Â = 1Â Â Â

Â Â â‡’Â AB = 10 m.Â (1)

In theÂ Î”Â ABD,

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â = tan 60Â°

â‡’Â Â Â Â Â Â Â Â Â

Substituting for AB from (1)

â‡’Â Â Â Â Â Â Â Â h + 10 =10Â

â‡’Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â h = 10(-1) = 7.32 m

Again , in theÂ Â Î”Â ABD,

Â = cos 60Â°

â‡’Â Â Â Â Â Â Â Â Â

â‡’Â Â Â x = 20 m.

Â

Â

Two men are on diametrically opposite sides of a tower. They measure the angles of elevation of the top of the tower as 20Â° and 24Â° respectively. If the height of the tower is 40m, find the distance between the two men.Â

In theÂ Î”ABD,

Â Â Â Â Â Â Â Â Â cot 24Â° =Â

â‡’Â Â Â Â Â Â Â Â Â 2.246 =Â

â‡’Â Â Â Â Â Â Â Â Â Â Â Â Â AB = 40 x 2.246 = 89.84 m

In theÂ Î”Â CBD,

Â Â Â Â Â Â Â cot 20Â° =Â

â‡’Â Â Â Â Â Â Â 2.747 =Â

â‡’Â Â Â Â Â Â Â Â Â Â Â BC = 40Â Ã—Â 2.747 = 109.88 m

The distance between the two men

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â = AC = AB + BC

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â = 89.84 + 109.88

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â = 199.72 m.

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