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?
To find the height of the pole, we can use trigonometry. Let's denote the height of the pole as h.
First, let's draw a diagram to visualize the problem.
```
/|
/ |
h / |
/ |
/ |
/_____|
16m
```
In the diagram, we have a right-angled triangle formed by the height of the pole, the length of its shadow, and the slope.
We know that the angle of elevation from the tip of the shadow to the sun is 20 degrees. This means that the angle between the shadow and the horizontal line is also 20 degrees.
Since the angle of the slope is 12 degrees, the angle between the shadow and the slope is equal to (20 - 12) = 8 degrees.
Now, using trigonometry, we can find the height of the pole.
In a right-angled triangle, 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 8 degrees is equal to the height of the pole (h) divided by the length of the shadow (16 meters).
Thus, we have:
tan(8 degrees) = h / 16
To find the value of h, we can rearrange the equation:
h = tan(8 degrees) * 16
Using a calculator, we can evaluate the right-hand side of the equation:
h ≈ 0.1405 * 16
h ≈ 2.248 meters
Therefore, the height of the pole is approximately 2.248 meters.
To find the height of the pole, we can use trigonometry. Let's break down the problem into smaller parts and use the given information to find the solution.
Step 1: Draw a diagram
Draw a diagram to represent the scenario. Label the relevant angles and lengths as described in the problem. This will help us visualize the problem better and understand how to solve it.
Step 2: Identify the known information
From the problem, we know the following:
- The slope makes an angle of 12∘ with the horizontal.
- The length of the shadow is 16 meters.
- The angle of elevation from the tip of the shadow to the sun is 20∘.
Step 3: Determine the relevant trigonometric ratios
To solve this problem, we can use the trigonometric ratio of tangent (tan). In this case, tan(angle) = opposite/adjacent.
We can apply this to the angles we have:
- For the angle of elevation from the tip of the shadow to the sun (20∘), we can use the tangent to relate the opposite side (height of the pole) and the adjacent side (length of the shadow).
- For the angle of the slope (12∘), we can use the tangent to relate the opposite side (length of the shadow) and the adjacent side (distance along the slope).
Step 4: Calculate the height of the pole
Let's use the trigonometric ratios to find the height of the pole.
Using the angle of elevation (20∘), we can set up the equation:
tan(20∘) = height of the pole / length of the shadow
Rearranging the equation, we get:
height of the pole = length of the shadow × tan(20∘)
Calculating this using the given values:
height of the pole = 16 × tan(20∘)
height of the pole ≈ 5.77 meters
Therefore, the height of the pole is approximately 5.77 meters.
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.
Your answer in total will be 2.3696