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.
Solution:- *PQ be the height of the boy
*AB be the height of the building
Given,
*Height of the boy = PQ = CB = 1.5 m
*Height of the building = AB = 30 m
∴ AC = AB - CB
∴ AC = 30 - 1.5
∴ AC = 28.5 m ------- (i)
*Angle of elevation from starting point Q to the top of the building = ∠APC = 30°
*Angle of elevation from final point S to the top of the building = ∠ARC = 60°
➙ In ∆APC, ∠ACP = 90°,
∴ tanP =
Opposite side of ∠P
/
Adjacent side of ∠P
∴ tan30° =
AC
/
PC
∴ tan30° =
28.5
/
PC
{ from (i) } ∴
1
/
√3
=
28.5
/
PC
{ ∵ tan30° = 1/√3 } ∴ PC = 28.5√3 ------- (ii)
➙ In ∆ARC, ∠ACR = 90°,
∴ tanR =
Opposite side of ∠R
/
Adjacent side of ∠R
∴ tan60° =
AC
/
RC
∴ tan60° =
28.5
/
RC
{ from (i) } ∴ √3 =
28.5
/
RC
{ ∵ tan60° = √3 } ∴ RC =
28.5
/
√3
*Multiplying Numerator and Denominator by √3,
∴ RC =
28.5
/
√3
×
√3
/
√3
∴ RC =
28.5√3
/
3
∴ RC = 9.5√3 ------- (iii)
Now,
*The distance walked by the boy towards the building
= QS
= PR
= PC - RC
= 28.5√3 - 9.5√3 { from (i), (ii) }
= 19√3
Hence,
The distance walked by the boy towards the building is 19√3 m.
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Try This..
✒ A kite is flying at a height of 60 m. The angle of elevation is 30°. Find the length of the string.
✒ The angle of elevation to the top of a building from two points on the ground is 30° and 60°. Find the distance between the two points.
✒ A man standing 40 m away from a tower sees the top of the tower at 45°. Find the height of the tower.
❌ Common Mistakes
▸ Forgetting to subtract the height of the observer (like the 1.5 m of the boy here).▸ Using wrong trigonometric values (like confusing tan 30° and tan 60°).Skipping diagrams:▸ A rough sketch can save you from silly errors.
📝 Related Questions:-
- Ex 9.1 Q1 - A circus artist is climbing a 20 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole, if the angle made by the rope with the ground level is 30°.
- Ex 9.1 Q2 - A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle 30° with it. The distance between the foot of the tree to the point where the top touches the ground is 8 m. Find the height of the tree.
- 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 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.
Queries Solved:-
Class 10 Ex 9.1
Ex 9.1 Q6 Class 10
Class 10 Ex 9.1 Q6
Class 10 Chap 9 Ex 9.1 Q6
Class 10 Some Applications Of Trigonometry
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