A pole at a right angle to the horizontal is located on a slope that makes an angle of 12 degrees with the horizontal. The pole's shadow is 16 meters long and points up the slope. The angle of elevation from the tip of the shadow to the sun is 20 degrees.
What is the height of the pole?
To determine the height of the pole, we can use trigonometric ratios. Let's break down the problem and find the solution step-by-step.
Step 1: Draw a diagram
Draw a diagram based on the given information to visualize the situation.
```
/|
/ | height (h)
/ |
/ |
/ |
/ |
/ |
/ 12° | 20°
/ |
/ | 16 meters
/ |
/ |
/ |
/____________|_________
distance to shadow
the sun
```
Step 2: Identify the right triangle
From the given information, we can identify a right triangle within the larger triangle formed by the pole, the distance to the sun, and the pole's shadow.
```
/|
/ | height (h)
/ |
/ |
/ |
/ |
/ |
/ |
/ |
L 12° | R
/________|_________
distance to shadow
the sun
```
Step 3: Determine the sides of the right triangle
In the right triangle, we have two known sides: the length of the shadow (16 meters) and the angle of elevation from the tip of the shadow to the sun (20 degrees).
Step 4: Use trigonometric functions
To find the height of the pole (h), we can use the tangent function:
tan(20°) = h / 16
Rearranging the equation, we get:
h = tan(20°) * 16
Step 5: Calculate the height of the pole
Now, we apply the equation to find the height of the pole:
h = tan(20°) * 16 = 5.671 meters (approx.)
Therefore, the height of the pole is approximately 5.671 meters.
You have two angles in the triangle (draw the figure), and one side. With two angles, you can figure the last angle.
I would use the law of sines, but you could use the law of cosines.