Richard is standing between two buildings in a town house development.The building on the left is 9 m away and the angle of elevation to its security spotlight A is 68 degrees. The building on the right is 6m away and the angle of elevation to its security spotlight B is 73 degrees which spotlight is farther away from ricchard and by how much ?
I hope you made a sketch
For the left triangle
cos68° = 9/hypotenuse
hypotenuse = 9/cos68 = 24.03
Calculate the hypotenuse for the right triangle in the same way, then compare those lengths.
regregreg
Well, Richard certainly has some interesting neighbors with their security spotlights. Let's do a little geometry and find out which spotlight is farther away from him.
First, let's calculate the height of the building on the left. We'll use a little trigonometry. Since we know the distance (9m) and the angle of elevation (68 degrees), we can use the tangent function to find the height.
tan(68 degrees) = height / 9m
height = 9m * tan(68 degrees) ≈ 24.71m
So, the building on the left is approximately 24.71m tall.
Next, let's calculate the height of the building on the right. Using the same approach, we have:
tan(73 degrees) = height / 6m
height = 6m * tan(73 degrees) ≈ 19.96m
So, the building on the right is approximately 19.96m tall.
Now, let's see which spotlight is farther away. Since both buildings have different heights, we can't just compare the distances. Instead, we'll compare the angles of depression from Richard's position to each spotlight.
The angle of depression from Richard to spotlight A (left building) is simply the complement of the angle of elevation, which is 90 - 68 = 22 degrees.
Similarly, the angle of depression from Richard to spotlight B (right building) is 90 - 73 = 17 degrees.
Now, we can clearly see that the angle of depression to spotlight B (17 degrees) is smaller than the angle of depression to spotlight A (22 degrees). So, spotlight B is closer to Richard.
To find out how much closer, we can subtract the angles:
22 degrees - 17 degrees = 5 degrees
Therefore, spotlight B is 5 degrees closer to Richard than spotlight A.
I hope that helps! Just remember, even in the world of security spotlights, it's all about the angles!
To find out which spotlight is farther away from Richard and by how much, we can use trigonometry. Let's denote the distance from Richard to spotlight A as x1 and the distance from Richard to spotlight B as x2.
First, we will find x1 using the angle of elevation and the given distance to the building on the left. We can use the tangent function, which relates the opposite side to the adjacent side of a right triangle:
tan(68 degrees) = x1 / 9 m
To solve for x1, we multiply both sides by 9 m and then take the tangent of 68 degrees:
x1 = tan(68 degrees) * 9 m
Next, we will find x2 using the angle of elevation and the given distance to the building on the right:
tan(73 degrees) = x2 / 6 m
To solve for x2, we follow the same steps:
x2 = tan(73 degrees) * 6 m
Now we can compare x1 and x2 to determine which spotlight is farther away from Richard. If x1 is greater than x2, then spotlight A is farther away. If x2 is greater than x1, then spotlight B is farther away.
Additionally, to find the difference in distances between the two spotlights, we can subtract the smaller distance from the larger distance (|x2 - x1|).