a woman looking out from a window of a building at a height of 30meters observed that the angle of depression of a flag pole was 44°. if the foot of the pole is 25meters from the foot of the building from the same horizontal, find (a) correct to the nearest whole number the angle of depression of the foot of the pole from the woman (b) the height of the flag pole.
The key is a good diagram.
On mine I labeled the window position as A and the base of the building as B, so AB = 30
I labeled the top of the flag pole as P, and its base as Q
So BQ = 25
We can find PQ by
PQ^2 = 30^2 + 25^2
PQ = ....
No look at triangle AQP,
angle APQ = 90+44 = 134°
and AQ we found above.
Back to triangle ABQ,
tan(angle AQB = 30/25
so we can find angle AQB = .....
and angle PQA = 90° - angle BQA = ....
allowing us to find angle QAP
now we can use the sine law in triangle AQP
PQ/ sin (QAP) = AQ/sin 134°
take over with the calculations
Draw a diagram. It should be clear that
(a) tanθ = 30/25
(b) (30-h)/25 = tan44°
seems to me that
AQ^2 = 30^2 + 25^2
Steve, thanks for the catch.
To solve this problem, we can use trigonometry, specifically the concept of angles of depression.
a) To find the angle of depression of the foot of the pole from the woman, we need to find the angle between the line of sight from the woman to the foot of the pole and the horizontal.
Let's assume the angle of depression of the foot of the pole from the woman is θ degrees. Since the angle of depression of the flag pole is given as 44°, we can use the fact that the angle of depression and angle of elevation are congruent when looking at the same object.
Therefore, θ = 44°.
b) To find the height of the flag pole, we can use the tangent function.
Let h be the height of the flag pole.
In the triangle formed by the woman, the foot of the pole, and the top of the pole, the opposite side is the height of the pole (h), the adjacent side is the horizontal distance between the foot of the pole and the woman (25m), and the angle θ is 44°.
Using the tangent function:
tan θ = opposite/adjacent
tan 44° = h/25
Rearranging the equation to solve for h:
h = tan 44° * 25
Using a calculator, the approximate value of h is 29.47 meters.
Therefore, the height of the flag pole is approximately 29.47 meters.