Rohit is standing at the top of the building observes a car at an angle of 30°, which is approaching the foot of the building at a uniform speed. 6 seconds later, the angle of depression of the car formed to be 60°, whose distance at that instant from the building is 25 m.
i. Find the height of the building.
ii. Find the distance between two positions of the car.
OR
Calculate total time is taken by the car to reach the foot of the building from the starting point.
iii. Find the distance of the observer from the car when it makes an angle of 60°
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Answer:
Let's solve this problem step by step:
i. Find the height of the building:
We can set up a right triangle to represent the situation. The building's height is one leg of the triangle, the distance from Rohit to the car when it's at the starting point is the other leg, and the distance from Rohit to the car when it's 25m away is the hypotenuse.
Let h be the height of the building, x be the initial distance from Rohit to the car, and 25m be the final distance when the angle is 60°.
Using trigonometry, we can write:
tan(30°) = h / x
tan(60°) = h / (x + 25)
We can rearrange the first equation to solve for h:
h = x * tan(30°)
Now, let's find the value of x from the second equation:
tan(60°) = h / (x + 25)
h = x * tan(30°)
tan(60°) = (x * tan(30°)) / (x + 25)
Solve for x:
x = 25 * (tan(60°) / tan(30°))
x = 25 * (√3)
x = 25√3
Now, we can find the height of the building:
h = x * tan(30°)
h = (25√3) * (√3 / 3)
h = 25 meters
So, the height of the building is 25 meters.
ii. Find the distance between two positions of the car:
The initial distance from Rohit to the car is x = 25√3 meters, and the final distance is 25 meters. The distance between these two positions of the car is:
Distance = Final Distance - Initial Distance
Distance = 25 meters - 25√3 meters
iii. Calculate the total time taken by the car to reach the foot of the building from the starting point:
To find the total time taken, we need to calculate the time it takes for the car to travel from its initial position to the final position. We know the car is approaching at a uniform speed, and the distance is decreasing by 25√3 - 25 meters. Let's call this distance "d."
d = 25√3 - 25 meters
Now, we can calculate the time it takes using the formula:
Time = Distance / Speed
Since the car is approaching the building, we need to use the relative speed. We have the angles and the fact that it's moving at a uniform speed, so the time taken will be the same at both positions.
We can use the relative speed formula as follows:
Relative Speed = (25√3 - 25) / 6 seconds
Now, you can calculate the total time it takes for the car to reach the foot of the building from the starting point using this relative speed.