A flagpole at right angle to the horizontal is located on a slope that makes an angle of 12∘ with the horizontal. The pole's shadow is 16 meters long and points directly down the slope. The angle of elevation from the tip of the shadow to the sun is 20∘. What is the height of the pole?
Draw a diagram. Label it
T = top of pole
B = bottom of pole
S = tip of shadow
In ∆SBT,
∠S = 8°
∠B = 102°
So, ∠T = 70°
Now, using the law of sines, the pole's height, BT can be found using
BT/sin8° = 16/sin70°
This assumes that the angle of elevation is measured from the horizontal, not from the slope of the ground.
2.3696
Well, well, well! Looks like we have some pole-related trigonometry here. Let's solve it, shall we?
First things first, we need to break down the problem. We have a right-angled triangle formed by the flagpole, its shadow, and the slope.
So, let's call the height of the flagpole "h", and the length of the slope "s". We're given that the shadow is 16 meters long, so we have the equation:
tan(12°) = h/16 (Equation 1)
Now, we're also told that the angle of elevation from the tip of the shadow to the sun is 20°. To find the height of the pole, we need the length of the slope, which can be determined by using the angle of elevation and the shadow length.
We know that the tangent of an angle equals the opposite side divided by the adjacent side. In this case, the opposite side is the height of the pole, and the adjacent side is the length of the slope.
So, we have another equation:
tan(20°) = h/s (Equation 2)
Now, we have two equations with two variables. Let's use them to find the height of the pole, shall we?
Using Equation 1, we can rearrange it to solve for "h":
h = 16 * tan(12°)
Plugging that value of "h" into Equation 2, we have:
16 * tan(12°) / s = tan(20°)
Now, let's solve for "s", the length of the slope:
s = 16 * tan(12°) / tan(20°)
Substituting the values into our equation:
s ≈ 16 * 0.2126 / 0.3639
After doing some calculations, we get:
s ≈ 9.34 meters
And voila, we have the length of the slope!
Now, using that result, we can go back to Equation 1 and solve for "h":
h = 16 * tan(12°)
Plugging in the values:
h ≈ 16 * 0.2126
After some final calculation, we find:
h ≈ 3.40 meters
So, the height of the pole is approximately 3.40 meters.
I hope I was able to shine some light-heartedness on this problem for you!
To find the height of the pole, we can use trigonometry. Let's label the angle of elevation from the base of the pole to the tip of the pole as A, the angle of elevation from the tip of the shadow to the sun as B, and the angle of the slope as C.
We are given that angle B is 20∘, so angle A is also 20∘ (since angle A and angle B are corresponding angles).
Now, we can use basic trigonometry ratios. The tangent function will be useful here.
The tangent of angle C is defined as the opposite side (the height of the pole) divided by the adjacent side (the length of the shadow). Therefore, we have:
tan(C) = height of the pole / length of the shadow
⇒ tan(12∘) = height of the pole / 16 meters
We can rearrange this equation to solve for the height of the pole:
height of the pole = 16 meters * tan(12∘)
Using a calculator, we can find the value of tan(12∘)≈0.211 of height of the pole:
height of the pole ≈ 16 meters * 0.211
height of the pole ≈ 3.376 meters
Therefore, the height of the pole is approximately 3.376 meters.
To find the height of the flagpole, we can use the properties of similar triangles.
Let's consider a diagram to understand the situation:
/|
/ |
h / |
/ |
/ | 20° (angle of elevation)
/____|
16m x
From the given information, we know that the angle of the slope is 12°, and the angle of elevation is 20°.
We need to find the value of 'h,' which represents the height of the pole.
To solve the problem, we can use the tangent function. The tangent of an angle is equal to the ratio of the opposite side to the adjacent side.
In this case, the tangent of the angle of elevation (20°) is equal to the ratio of the height of the pole ('h') to the length of the shadow (16m).
By applying the tangent function, we have:
tan(20°) = h / 16m
To find the value of 'h', we can rearrange the equation:
h = tan(20°) * 16m
Now, we can calculate 'h':
Using a calculator:
h ≈ 5.759 meters (rounded to three decimal places)
So, the height of the flagpole is approximately 5.759 meters.