From a balloon vertically above a straight road, the angles of depression of two cars at an instant are found to be 45° and 60°. If the cars are 100 m apart, find the height of the balloon

Solution:

Consider the height of the balloon at P be h meters.

Consider A and B as the two cars.

So AB = 100 m.

In ∆PAQ,

AQ = PQ = h

In ∆PBQ,

PQ/BQ = tan 60° = √3

From the figure

h/ (h - 100) = √3

By cross multiplication

h = √3(h - 100)

So we get

h = 100√3/ (√3 - 1)

Let us multiply and divide by (√3 + 1)

= 100√3/ (√3 - 1) x (√3 + 1)/ (√3 + 1)

By further calculation

= (100 x 3 + 100 √3)/ (3 - 1)

= 100 (3 + √3)/ 2

So we get

= 50(3 + √3) m

Therefore, the height of the balloon is 50(3 + √3) m.

 ✦ Try This: From the top of a 50 m high tower, the angles of depression of the top and bottom of a pole are observed to be 45° and 60° respectively. Find the height of the pole.

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

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NCERT Exemplar Class 10 Maths Exercise 8.4 Sample Problem 2

From a balloon vertically above a straight road, the angles of depression of two cars at an instant are found to be 45° and 60°. If the cars are 100 m apart, find the height of the balloon

Summary:

From a balloon vertically above a straight road, the angles of depression of two cars at an instant are found to be 45° and 60°. If the cars are 100 m apart, the height of the balloon is 50 (3 + √3) m

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