A boy and a girl are two friends from the same college and they were studying in ornithology. It is a branch of zoology which deals with the study of birds. Now they observed the elevation of birds, their position and different actions of birds.
(1) What is the height of the bird flying from the ground level?
(ii) A boy standing on a horizontal plane, find a bird flying at a distance of 200 m from him at an elevation of 30°. A girl standing on the root of a 50 m high building, if the angle of elevation of the same bird to be 45degree. A boy and a girl are an opposite sides of the bird. Find the distance of the bird from girl.
To determine the height of the bird flying from the ground level and the distance of the bird from the girl, we can use trigonometry.
First, let's calculate the height of the bird from the ground level.
We have the boy's position on a horizontal plane and the angle of elevation to the bird at 30°. Since we know the distance of the bird from the boy is 200m, we can use the tangent function.
1) tan(30°) = height of the bird / distance to the bird
tan(30°) = h / 200 (where h is the height of the bird)
h = 200 * tan(30°)
Next, let's calculate the distance of the bird from the girl.
We have the girl's position on the root of a 50m high building and the angle of elevation to the bird at 45°. Since the height of the building is 50m, we can consider the total height from the ground level to the bird when calculating the distance.
2) tan(45°) = (height of the bird + height of the building) / distance to the bird
tan(45°) = (h + 50) / d (where d is the distance of the bird from the girl)
d * tan(45°) = h + 50
d = (h + 50) / tan(45°)
Now, let's substitute the value of h from equation (1) into equation (2).
d = (200 * tan(30°) + 50) / tan(45°)
Simplifying, we can calculate the value of d.
To find the height of the bird flying from the ground level, we can use trigonometry and the concept of angle of elevation.
First, let's work on finding the height of the bird from the boy's perspective. Given that the bird is flying at a distance of 200 m from the boy and at an elevation of 30 degrees, we can use the tangent function.
Using the trigonometric identity:
tangent(angle) = opposite/adjacent
tangent(30 degrees) = height of bird / distance to bird
By rearranging the formula, we can solve for the height of the bird:
height of bird = distance to bird * tangent(30 degrees)
height of bird = 200 * tan(30 degrees)
Now, calculate the height of the bird from the boy's perspective:
height of bird = 200 * tan(30 degrees)
height of bird ≈ 115.47 m
Next, let's find the distance of the bird from the girl. Given that the girl is standing on the roof of a 50 m high building and the angle of elevation of the bird from her perspective is 45 degrees, we can use trigonometry again.
Since the height of the building is known, and the angle of elevation is given, we can use the tangent function to find the distance.
Using the trigonometric identity:
tangent(angle) = opposite / adjacent
tangent(45 degrees) = height of bird from girl / distance to bird from girl
By rearranging the formula, we can solve for the distance to the bird from the girl:
distance to bird from girl = height of bird from girl / tangent(45 degrees)
distance to bird from girl = 50 / tan(45 degrees)
Now, calculate the distance of the bird from the girl:
distance to bird from girl = 50 / tan(45 degrees)
distance to bird from girl ≈ 50 m
Therefore, the distance of the bird from the girl is approximately 50 meters.
Height of bird ---- h
according to my sketch:
h/200 = sin30°
h = 200sin30 = 100
So the vertical height of bird above the girl is 50 m
let the distance between girl and bird be x
sin 45° = 50/x
x = 50/sin45 = appr 70.7 m