Ex 9.1 Q7 - From a point on the ground, the angles of elevation of the bottom and the top of a transmission tower fixed at the top of a 20 m high building are 45° and 60° respectively. Find the height of the tower.
Solution:- *CB be the height of the building
*AC be the height of the tower
*D be the point of observation
Given,
*Height of the building = CB = 20 m
*Angle of elevation to the bottom of the tower from point D = ∠CDB = 45°
*Angle of elevation to the top of the tower from point D = ∠ADB = 60°
➙ In ∆CDB, ∠CBD = 90°,
∴ tanD =
Opposite side of ∠D
/
Adjacent side of ∠D
∴ tan45° =
CB
/
DB
∴ tan45° =
20
/
DB
{ Given } ∴ 1 =
20
/
DB
{ ∵ tan45° = 1 } ∴ DB = 20 ------- (i)
➙ In ∆ADB, ∠ABD = 90°,
∴ tanD =
Opposite side of ∠D
/
Adjacent side of ∠D
∴ tan60° =
AB
/
DB
∴ tan60° =
AB
/
20
{ from (i) } ∴ √3 =
AB
/
20
{ ∵ tan60° = √3 } ∴ AB = 20√3
Now,
*Height of the tower
= AC
= AB - CB
= 20√3 - 20
= 20(√3 - 1)
Hence,
The height of the tower is 20(√3 - 1) m.
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Try This..
✒ A person observes the top of a tower at an angle of 45°. If the tower is 30 m high, find the distance of the person from the base of the tower.
✒ Two towers of height 20 m and 30 m are 100 m apart. Find the angle of elevation of the top of the taller tower from the top of the shorter tower.
✒ A ladder leaning against a wall makes an angle of 60° with the ground. If the foot of the ladder is 5 m away from the wall, find the height at which the ladder touches the wall.
✒ From a point on the ground, the angles of elevation of the top of a building and a flagstaff on top of the building are 45° and 60°. If the height of the building is 25 m, find the height of the flagstaff.
❌ Common Mistakes
▸ Mixing up opposite and adjacent sides in the triangle.
▸ Forgetting to subtract the height of the building when calculating just the tower's height.
▸ Using incorrect values for tan(45°) and tan(60°). Remember: tan(45°) = 1; tan(60°) = √3
▸ Misplacing values in the triangle and confusing which angle goes where.
▸ Not labeling your diagram properly (this leads to wrong assumptions).
📝 Related Questions:-
- Ex 9.1 Q3 - A contractor plans to install two slides for the children to play in a park. For the children below the age of 5 years, she prefers to have a slide whose top is at a height of 1.5 m, and is inclined at an angle of 30° to the ground, whereas for elder children, she wants to have a steep slide at a height of 3 m, and inclined at an angle of 60° to the ground. What should be the length of the slide in each case?
- Ex 9.1 Q4 - The angle of elevation of the top of a tower from a point on the ground, which is 30 m away from the foot of the tower, is 30°. Find the height of the tower.
- Ex 9.1 Q5 - A kite is flying at a height of 60 m above the ground. The string attached to the kite is temporarily tied to a point on the ground. The inclination of the string with the ground is 60°. Find the length of the string, assuming that there is no slack in the string.
- Ex 9.1 Q6 - A 1.5 m tall boy is standing at some distance from a 30 m tall building. The angle of elevation from his eyes to the top of the building increases from 30° to 60° as he walks towards the building. Find the distance he walked towards the building.
- Ex 9.1 Q8 - A statue, 1.6 m tall, stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is 60° and from the same point the angle of elevation of the top of the pedestal is 45°. Find the height of the pedestal.
Queries Solved:-
Class 10 Ex 9.1
Ex 9.1 Q7 Class 10
Class 10 Ex 9.1 Q7
Class 10 Chap 9 Ex 9.1 Q7
Class 10 Some Applications Of Trigonometry
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