The angle of elevation of the bottom of a window 10 meter above the ground level from a point on the ground is 30degrees.
A pole projecting outwards and upwards from the bottom of the window makes an angle of 30 degrees with the wall.
If the angle of elevation of the top of the pole observed from the same point on the ground is 60 degrees, what is the the length of the pole?
Ah, such carefully contrived angles.
Label the
point on the ground Q
base of the wall A
bottom of the window B
top of the pole P
∆ABQ is a 30-60-90 ∆, so BQ=20
∆BPQ is also 30-60-90, so BP=20/√3
so, the pole is 20/√3 = 11.5m
To find the length of the pole, we can use trigonometric ratios like sine, cosine, and tangent. Let's break down the problem and go step by step:
Step 1: Set up the problem
Draw a picture representing the scenario. Identify the relevant angles and sides.
In this case, we have a right triangle formed by the ground, the top of the pole, and the point of observation on the ground.
Step 2: Identify the known information
We are given:
- The angle of elevation from the point on the ground to the bottom of the window is 30 degrees.
- The length of the window is 10 meters.
- The angle between the pole and the wall is 30 degrees.
- The angle of elevation from the point on the ground to the top of the pole is 60 degrees.
Step 3: Determine the length of the pole
Let's find the length of the pole using trigonometric ratios. Since we know the length of the window and the angle of elevation of its bottom, we can determine the distance from the point on the ground to the bottom of the pole.
Using the sine ratio, where sine(theta) = opposite/hypotenuse:
sin(30) = (height of the window) / (distance to the bottom of the window)
sin(30) = 10 / x (let x represent the distance to the bottom of the window)
Solving for x:
x = 10 / sin(30)
x = 20 meters (approx.)
Hence, the distance from the point on the ground to the bottom of the pole is approximately 20 meters.
Now, to find the length of the pole, we need to find the length between the bottom of the window and the top of the pole. We will use the tangent ratio, where tangent(theta) = opposite/adjacent.
In the triangle formed by the bottom of the window, the top of the pole, and the distance to the top of the pole, we have:
tan(60) = length of the pole / 20
Solving for the length of the pole:
length of the pole = tan(60) * 20
length of the pole ≈ 34.64 meters
Therefore, the length of the pole is approximately 34.64 meters.