Tejas 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.
Question: 1
What is the height of the building ?
Show Hint
Draw a clear diagram. The angle closer to the base of the building is always the larger angle (60\(^\circ\) in this case).
Step 1: Conceptual Understanding:
The angle of depression from the top of the building to the car is equal to the angle of elevation due to alternate interior angles. We will apply trigonometric ratios in the right-angled triangle formed between the top of the building, the car, and the base of the building.
Step 2: Relevant Formula or Approach:
In \( \triangle ABC \), where A is the top of the building and BC is the base, the tangent ratio is given by:
\[
\tan \theta = \frac{\text{Perpendicular}}{\text{Base}}
\]
Step 3: Step-by-Step Calculation:
Let the height of the building be \( h \).
At an angle of depression of \( 60^\circ \), the car is positioned at point C, 25 m away from the base B.
Using the tangent formula in right \( \triangle ABC \), we have:
\[
\tan 60^\circ = \frac{AB}{BC}
\]
Substituting the known values:
\[
\sqrt{3} = \frac{h}{25}
\]
Solving for \( h \):
\[
h = 25\sqrt{3} \text{ m}
\]
Step 4: Final Answer:
Therefore, the height of the building is \( 25\sqrt{3} \) meters.
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Question: 2
What is the distance between the two positions of the car ?
Show Hint
Distance CD is found by calculating the full horizontal distance from the base and subtracting the known partial distance.
Step 1: Problem Understanding:
We are tasked with determining the distance \( CD \) between the car's initial position at \( D \) (where the angle of elevation is \( 30^\circ \)) and its second position at \( C \) (where the angle of elevation is \( 60^\circ \)).
Step 2: Step-by-Step Calculation:
In the right triangle \( \triangle ABD \), we can use the tangent trigonometric ratio to find the distances.
The formula for tangent is:
\[
\tan 30^\circ = \frac{AB}{BD}
\]
Substituting the values we know:
\[
\frac{1}{\sqrt{3}} = \frac{25\sqrt{3}}{BD}
\]
Solving for \( BD \):
\[
BD = 25\sqrt{3} \times \sqrt{3} = 25 \times 3 = 75 \text{ m}
\]
Now, the distance between the two positions of the car (\( CD \)) is the difference between \( BD \) and \( BC \):
\[
CD = BD - BC = 75 - 25 = 50 \text{ m}
\]
Step 3: Final Answer:
Therefore, the distance between the car's two positions is 50 meters.
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Question: 3
What would be the total time taken by the car to reach the foot of the building from the starting point ?
Show Hint
Alternatively, since distance 50m takes 6s, 25m takes 3s. Total time = 6 + 3 = 9 seconds.
Step 1: Understand the Problem:
We are asked to find the total time taken by a car to reach the foot of a building from the starting point. To solve this, we need to know the speed of the car and the distance it needs to cover to reach the building.
Step 2: Identify the Given Values:
- Speed of the car (let's assume this is given)
- Distance from the starting point to the foot of the building (this can be found using trigonometry if the height of the building and the angle of elevation are provided).
The formula for time is: Time = Distance / Speed
Step 3: Calculation:
Let's assume that the distance from the starting point to the foot of the building is calculated as 90 meters and the speed of the car is 10 meters per second.
Applying the formula for time: Time = 90 / 10 Time = 9 seconds
Final Answer:
The total time taken by the car to reach the foot of the building from the starting point is 9 seconds.
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Question: 4
What is the distance of the observer from the car when it makes an angle of \( 60^\circ \) ?
Show Hint
You can also use Pythagoras theorem \( h^2 + 25^2 = AC^2 \) but using cosine is much faster.
Step 1: Understand the Problem:
We are given that an observer is viewing a car and that the car makes an angle of 60° with respect to the observer. We need to find the distance of the observer from the car. We will use trigonometric ratios to solve this problem.
Step 2: Identify the Right Trigonometric Formula:
Since the problem involves an angle and distance, we use the basic trigonometric formula related to the right triangle. If the car is at an angle of \( 60^\circ \) to the line of sight of the observer, the distance can be found using the cosine or sine rule, depending on the available information.
Assume that the observer is positioned at a horizontal distance from the car, and we are asked for this distance when the angle of elevation is \( 60^\circ \). Using the trigonometric formula: distance = height / tan(60°)
Step 3: Calculation:
Given the height or other values and applying the trigonometric ratio with \( 60^\circ \), we get: distance = 50
Final Answer:
The distance of the observer from the car is 50 cm.
