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
a. d1 = 200/sin43 = 293 Ft.
b. d2 = 200/tan43.
c. d3 = d2/Cos16.
d. h = 200 + d2*Tan16
To solve this problem, we can use trigonometric ratios such as sine, cosine, and tangent. Let's break down each part of the problem:
a) To find the distance from the top of the building to the man (line of sight), we can use the tangent function. The tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is the height of the building (200 ft) and the angle is the angle of depression (43°). So we have:
tangent(43°) = opposite/adjacent
tangent(43°) = 200 ft / x (where x is the distance we're looking for)
Rearranging the equation, we have:
x = 200 ft / tangent(43°)
b) To find the distance from the top of the building to the bird/man's location (horizontal distance), we'll use the sine function. The sine of an angle is equal to the opposite side divided by the hypotenuse. In this case, the angle is the angle of elevation (16°) and the opposite side is the height of the building. So we have:
sine(16°) = opposite/hypotenuse
sine(16°) = 200 ft / y (where y is the distance we're looking for)
Rearranging the equation, we have:
y = 200 ft / sine(16°)
c) To find the distance from the top of the building to the bird (line of sight), we'll use the cosine function. The cosine of an angle is equal to the adjacent side divided by the hypotenuse. In this case, the angle is the angle of elevation (16°) and the adjacent side is the distance we found in part a. So we have:
cosine(16°) = adjacent/hypotenuse
cosine(16°) = x / z (where z is the distance we're looking for)
Rearranging the equation, we have:
z = x / cosine(16°)
d) To find the height of the bird flying, we'll subtract the height of the man from the height of the building. So we have:
Height of bird = height of building - height of man
Height of bird = 200 ft - x (where x is the distance we found in part a)
Simplifying the equation, we have:
Height of bird = 200 ft - (200 ft / tangent(43°))
Now you can substitute the values into the equations to find the distances and heights. Remember to use a calculator in degree mode to calculate the trigonometric functions.