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.
Solution:- *AB be the statue.
*BC be the pedestal.
*D be the point on ground from where
angle of elevation is measured.
To find: Height of pedestal i.e BC.
Given,
*Height of statue = AB = 1.6 m
*Angle of elevation of the top of
statue = ∠ADC = 60°
*Angle of elevation of the top of
pedestal = ∠BDC = 45°
*Since, the pedestal is perpendicular to the ground,
∴ ∠ACD = 90°
➙ In ∆BDC, ∠BCD = 90°
∴ tanD =
Opposite side of ∠D
/
Adjacent side of ∠D
∴ tan45° =
BC
/
CD
∴ 1 =
BC
/
CD
{ ∵ tan45° = 1 } ∴ CD = BC ------- (1)
➙ In ∆ADC, ∠ACD = 90°
∴ tanD =
Opposite side of ∠D
/
Adjacent side of ∠D
∴ tan60° =
AC
/
CD
∴ tan60° =
AC
/
BC
{ From (1) } ∴ tan60° =
AB + BC
/
BC
{ ∵ AC = AB + BC } ∴ tan60° =
1.6 + BC
/
BC
{ Given, AB = 1.6 } ∴ √3 =
1.6 + BC
/
BC
{ ∵ tan60° = √3 } ∴ √3BC = 1.6 + BC
∴ √3BC - BC = 1.6
∴ BC(√3 - 1) = 1.6
∴ BC =
1.6
/
√3 - 1
*Multiplying Numerator and Denominator by √3 + 1,
∴ BC =
1.6
/
√3 - 1
×
√3 + 1
/
√3 + 1
*Using formula, (a + b) (a - b) = a² - b²
=
1.6(√3 + 1)
/
(√3)² - (1)²
=
1.6(√3 + 1)
/
3 - 1
=
1.6(√3 + 1)
/
2
= 0.8(√3 + 1)
∴ BC = 0.8(√3 + 1)
Hence,
The height of the pedestal is 0.8(√3 + 1) m.
------------------------------------
Try This..
✒ The angle of elevation of a cloud from a point 120 m above a lake is 30°, and the angle of depression of its reflection in the lake is 60°. Find the height of the cloud.
✒ A tower 10 m high casts a shadow 10 m long. Find the angle of elevation of the Sun.
✒ A ladder is leaning against a wall making an angle of 60° with the ground. If the foot is 2.5 m away from the wall, find the length of the ladder.
❌ Common Mistakes
Wrong triangle selection:▸ Always match angles with the correct triangle (statue+pedestal vs pedestal alone).Not rationalizing denominators:▸ Especially when dealing with square roots like √3.Forgetting units:▸ Final answer should always include meters.Mixing up ‘from’ and ‘to’ in angle of elevation:▸ Visualizing the diagram helps eliminate this confusion.
📝 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.
Queries Solved:-
Class 10 Ex 9.1
Ex 9.1 Q8 Class 10
Class 10 Ex 9.1 Q8
Class 10 Chap 9 Ex 9.1 Q8
Class 10 Some Applications Of Trigonometry
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