From the roof of the building 25 meters high, the angle of elevation of the top of the pole is 21°11’ , from the bottom of the building, the angle of elevation is 43°2’, determine the height of the pole .
To determine the height of the pole, we can use trigonometry and the concept of similar triangles. Let's break down the problem step by step.
1. Draw a diagram: Start by visualizing the scenario and sketching a diagram of the situation. Draw a vertical line to represent the building, and another line perpendicular to it to represent the pole. Label the height of the building as 25 meters.
2. Determine the angles: From the top of the building, the angle of elevation to the top of the pole is 21°11' (or 21.1833°), and from the bottom of the building, the angle of elevation is 43°2' (or 43.0333°).
3. Identify the triangles: In this case, we have two right-angled triangles. The first triangle is formed by the building, the pole, and a horizontal line connecting the top of the pole to the bottom of the building. The second triangle is formed by the building, the pole, and a horizontal line connecting the bottom of the building to the top of the pole.
4. Solve the first triangle: In the first triangle, the opposite side relative to the angle of 21.1833° is the height of the pole (let's call it h), and the adjacent side is the height of the building (25 meters). You can use the tangent function to find h. The equation would be:
tan(21.1833°) = h / 25
Rearranging the equation:
h = 25 * tan(21.1833°)
Calculate h using a scientific calculator or trigonometric table.
5. Solve the second triangle: In the second triangle, the opposite side relative to the angle of 43.0333° is also the height of the pole (h), and the adjacent side is the height of the building plus the height of the pole (25 + h). Again, we can use the tangent function:
tan(43.0333°) = h / (25 + h)
Rearranging the equation:
h = (25 + h) * tan(43.0333°)
Solve this equation for h. You can do this by expanding the equation and simplifying it until you isolate h on one side.
6. Substitute the value of h: Once you find the value of h from the above equation, substitute it back into either equation in step 4 or 5 to find the actual height of the pole.
Following these steps will help you determine the height of the pole using the given angles of elevation from the top and bottom of the building.