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 ?
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 ?
We draw 2 rt triangles:
X1 = 9 m. = Hor side.
Y1 = Ht. of light(L1) = Ver. side.
Ae = 68 Deg. = Angle of elevation.
Z1 = Hyp. = X1/cosAe = 9 / cos68=24 m.=
Dist. from Rich. to L1.
X2 = 6 m. = Hor. side.
Y2 = Ht. of L2 = Ver side.
Be = 73 Deg. = Angle of elevation.
Z2 = Hyp. = X2/cosBe = 6 / cos73=20.5 m
Z1 - Z2 = 24 - 20.5 = 3.5 m.
Therefore, the spotlight on the left is 3.5 m farther away.
y
thanx
To determine which spotlight is farther away from Richard and by how much, we can use trigonometry.
Let's label the distances from Richard to each building as follows:
- Distance to the left building: D1 = 9m
- Distance to the right building: D2 = 6m
We also have the angles of elevation to each spotlight:
- Angle of elevation to spotlight A: θ1 = 68 degrees
- Angle of elevation to spotlight B: θ2 = 73 degrees
To find the distance to each spotlight, we can use the tangent function, which relates the angle of elevation to the distance:
tan(θ) = opposite/adjacent
For spotlight A:
tan(θ1) = D1/opposite1
opposite1 = D1/tan(θ1)
For spotlight B:
tan(θ2) = D2/opposite2
opposite2 = D2/tan(θ2)
Now, let's calculate the distances to each spotlight:
opposite1 = 9m/tan(68 degrees) ≈ 3.408m
opposite2 = 6m/tan(73 degrees) ≈ 1.556m
So, the distance to spotlight A is approximately 3.408m, and the distance to spotlight B is approximately 1.556m.
Since the distance to spotlight A is greater than the distance to spotlight B, we can conclude that spotlight A is farther away from Richard. The difference in distance between the two spotlights is:
Distance difference = Distance to spotlight A - Distance to spotlight B
= 3.408m - 1.556m
= 1.852m
Therefore, spotlight A is farther away from Richard by approximately 1.852 meters.