Richard is standing between two buildings in a townhouse development. The building on the left is 9m away and the angle of elevation to its security spotlight, A, is 68degree. The building on the right is 6m away and the angle of elevation to its security spotlight, B, is 73degree. Which spotlight is farther away from Richard, and by how much?
To determine which spotlight is farther away from Richard, we need to find the distance from Richard to each spotlight.
Let's first calculate the distance from Richard to spotlight A. We can use trigonometry here. Consider the triangle formed by Richard, the building on the left, and the spotlight A. The angle of elevation (α) is the angle between the line connecting Richard to the building and the line connecting Richard to the spotlight. In this case, the angle of elevation to spotlight A is 68 degrees.
Using the tangent function, we can set up the equation:
tan(α) = opposite/adjacent
tan(68) = opposite/9
opposite = 9 * tan(68)
opposite ≈ 28.28 meters
So, the distance from Richard to spotlight A is approximately 28.28 meters.
Now, let's calculate the distance from Richard to spotlight B. Similarly, consider the triangle formed by Richard, the building on the right, and the spotlight B. The angle of elevation to spotlight B is 73 degrees.
Using the same equation as before:
tan(α) = opposite/adjacent
tan(73) = opposite/6
opposite = 6 * tan(73)
opposite ≈ 18.75 meters
So, the distance from Richard to spotlight B is approximately 18.75 meters.
Comparing the distances, we see that spotlight A is farther away from Richard by approximately:
28.28 - 18.75 ≈ 9.53 meters
Therefore, spotlight A is farther away from Richard by approximately 9.53 meters.