The angle of elevation of the top of a vertical tower from a point on the ground is 60°. From another point 10 m vertically above the first, its angle of elevation is 45°. Find the height of the tower

Solution:

Given, the angle of elevation of the top of a vertical tower from a point on the ground is 60°

From another point vertically above the first, the angle of elevation is 45𝆩

We have to find the height of the tower.

Let AP be the vertical tower.

Height of the vertical tower, AP = 10m

Angle of elevation, ∠TPO = 60𝆩

Angle of elevation of another point, ∠TAB = 45𝆩

From the above diagram,

OB = AP = 10m

Let us consider,

Distance from the point of observation to the foot of the tower, PO = x m

Total height of the tower, TO = h m

From the figure, TB = (h - 10)m

Considering the ∠TPO, we get,

tan60𝆩 = TO/PO

Since, tan60° = √3

√3 = h/x

x = h/√3--------------(1)

Considering the ∠TAB, we get,

tan45° = TH/AB

tan45° = TO - TH/AB

Since, tan45° = 1,

1 = h - 10/AB

AB = h -10

x = h -10-----------------(2)

Comparing (1) and (2), we get,

h/√3 = h -10

h - h/√3 = 10

h(1 - 1√3) = 10

h(√3 - 1/√3) = 10

h = 10√3/√3 - 1

Taking conjugate,

h = 10√3/√3 - 1 × √3 - 1/√3 - 1

h = 30 + 10√3/3 - 1

h = 30 + 10√3/2

h = 15 + 5√3

h = 5(3 + √3)

h = 5√3 (√3 + 1)

Therefore, the height of the tower is 5√3 (√3 + 1)m.

✦ Try This: From a point on the ground, 20 m away from the foot of a vertical tower, the angle of elevation of the top of the tower is 60°, what is the height of the tower?

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 8

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NCERT Exemplar Class 10 Maths Exercise 8.4 Problem 16

The angle of elevation of the top of a vertical tower from a point on the ground is 60°. From another point 10 m vertically above the first, its angle of elevation is 45°. Find the height of the tower

Summary:

The angle of elevation of the top of a vertical tower from a point on the ground is 60° . From another point 10 m vertically above the first, its angle of elevation is 45𝆩. The height of the tower is 5√3 (√3 + 1)m

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