From the top of a 200 foot building, a bird is sighted overhead and a man is observed directly below the bird at ground level. The angle of elevation of the bird is 16*, and the angle of depression of the man is 43* find :
a) the distance from the top of the building to the man (line of sight)
b) the distance from the top of the building to the bird/man's location (horizontal distance)
c) the distance from the top of the building to the bird (line of sight)
d) the height of the bird flying
(Must show all work) any help would be appreciated!!
Draw the diagram. Label the angles and sides of the trangle. Use your basic trig relationships.
To solve this problem, we can use trigonometric ratios and properties of right triangles. Let's break it down step by step.
a) To find the distance from the top of the building to the man (line of sight), we can use the tangent function since we have the angle of depression.
The tangent function is defined as the opposite side divided by the adjacent side in a right triangle. In this case, the opposite side is the height of the building, 200 feet, and the adjacent side is the horizontal distance from the top of the building to the man. Let's call this distance x.
So, tan(43°) = 200/x
To solve for x, we can rearrange the equation: x = 200/tan(43°)
Using a calculator, we find x ≈ 200/0.931 = 214.88 feet
Therefore, the distance from the top of the building to the man (line of sight) is approximately 214.88 feet.
b) To find the distance from the top of the building to the bird/man's location (horizontal distance), we can again use the tangent function.
In this case, we know the angle of elevation of the bird is 16°, and we want to find the horizontal distance. Let's call this distance y.
So, tan(16°) = 200/y
To solve for y, we can rearrange the equation: y = 200/tan(16°)
Using a calculator, we find y ≈ 200/0.286 = 697.66 feet
Therefore, the distance from the top of the building to the bird/man's location (horizontal distance) is approximately 697.66 feet.
c) To find the distance from the top of the building to the bird (line of sight), we can use the Pythagorean theorem.
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
In this case, we know the horizontal distance is y ≈ 697.66 feet and the distance from the top of the building to the man is x ≈ 214.88 feet. Let's call the distance from the top of the building to the bird z.
Using the Pythagorean theorem, we have: z^2 = x^2 + y^2
z^2 = (214.88)^2 + (697.66)^2
Using a calculator, we find z ≈ √(214.88^2 + 697.66^2) = 731.81 feet
Therefore, the distance from the top of the building to the bird (line of sight) is approximately 731.81 feet.
d) To find the height of the bird flying, we can subtract the height of the building from the distance from the top of the building to the bird (line of sight).
The height of the bird flying = z - height of the building = 731.81 - 200 = 531.81 feet
Therefore, the height of the bird flying is approximately 531.81 feet.