This a problem I need help with. A vertical ple stands by a road that is inclined 10 degrees to the horizontal. When the angle of elevation of the sun is 23 degrees, the pole casts a shadow 38 ft long directly downhill along the road. How long is the pole?
Inclined 10 degrees to the horizontal? That is hardly vertical. But at any rate, you know the angle of the shadow(sunlight) to the horizontal, the length and the base angle.
ASA you can use the law of sines, which I recommend.
To find the length of the pole, we can use trigonometric ratios.
Let's consider the given information:
- The angle of inclination of the road to the horizontal is 10 degrees.
- The angle of elevation of the sun is 23 degrees.
- The length of the shadow cast by the pole directly downhill along the road is 38 ft.
We can draw a diagram to visualize the situation. The pole is represented by a vertical line, and the shadow is represented by a line extending from the bottom of the pole parallel to the road.
In the diagram, we have a right triangle formed by the pole, its shadow, and a line from the top of the pole to the sun. The angle between the line from the top of the pole to the sun and the pole itself is the angle of elevation, which is 23 degrees. The angle between the shadow and the road is the angle of inclination, which is 10 degrees.
Using trigonometric ratios, we can determine the length of the pole. Let's use the tangent function in this case.
tangent(angle) = opposite/adjacent
In this scenario:
tangent(23 degrees) = opposite (length of the pole)/adjacent (length of the shadow)
We know the length of the shadow is 38 ft, so we can rearrange the equation above to solve for the length of the pole:
length of the pole = tangent(23 degrees) * length of the shadow
Calculating the value:
length of the pole = tangent(23 degrees) * 38 ft
Using a scientific calculator or an online calculator:
length of the pole ≈ 0.4245 * 38 ft
length of the pole ≈ 16.119 ft
Therefore, the length of the pole is approximately 16.119 ft.