Heights and Distance – Case Based Questions

Q 11 :

Rohit is standing at the top of a building and observes a car at an angle of depression of $30 °$ as it approaches the base of the building at a uniform speed. 6 seconds later, the angle of depression increases to $60 °$ , and at that moment, the car is 25 m away from the building.

Based on the above information, answer the following questions:

(i) What is the height of the building?

$25 \sqrt{3} m$
$25 \sqrt{- 3} m$
$25 \sqrt{2} m$
$24 \sqrt{3} m$

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(1)

From the figure:

$∠ D A M = ∠ A D B = 30 ° and ∠ C A M = ∠ A C B = 60 ° (A l t e r n a t e a n g l e s) I n ∆ A B C t a n 60 ° = \frac{A B}{B C} \sqrt{3} = \frac{A B}{25} \Rightarrow A B = 25 \sqrt{3} H e n c e, Height of building = 25 \sqrt{3} m$

Q 12 :

Rohit is standing at the top of a building and observes a car at an angle of depression of $30 degree$ as it approaches the base of the building at a uniform speed. 6 seconds later, the angle of depression increases to $60 degree$ , and at that moment, the car is 25 m away from the building.

Based on the above information, answer the following questions:

(ii) What is the distance between the two positions of the car?

$50 m$
$- 50 m$
$45 m$
$30 m$

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(1)

$I n ∆ A B D : t a n 30 ° = \frac{A B}{B D} \frac{1}{\sqrt{3}} = \frac{25 \sqrt{3}}{B D} \Rightarrow B D = 75 m Distance between the two positions of the car : BD - BC = 75 - 25 = 50 m$

Q 13 :

Rohit is standing at the top of a building and observes a car at an angle of depression of $30 degree$ as it approaches the base of the building at a uniform speed. 6 seconds later, the angle of depression increases to $60 degree$ , and at that moment, the car is 25 m away from the building.

Based on the above information, answer the following questions:

(iii) (a) What would be the total time taken by the car to reach the foot of the building from starting point?

8 s
7 s
6 s
9 s

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(4)

Time taken to cover 50 m distance (i.e., DC): = 6s

Thus, time taken to cover 25 m (i.e., BC): = 3s

Total time from starting point to foot of building: $D C + B C = 6 + 3 = 9 s$

Q 14 :

Rohit is standing at the top of a building and observes a car at an angle of depression of $30 degree$ as it approaches the base of the building at a uniform speed. 6 seconds later, the angle of depression increases to $60 degree$ , and at that moment, the car is 25 m away from the building.

Based on the above information, answer the following questions:

(iii) (b) What is the distance of the observer from the car when it makes an angle of $60 ° ?$

$40 m$
$10 m$
$50 m$
$60 m$

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(3)

$I n ∆ A B C : c o s 60 ° = \frac{B C}{A C} \frac{1}{2} = \frac{25}{A C} A C = 50 m Hence, Distance of the observer from the car = 50 m$

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