Mike's hot-air balloon is 875.0 m directly above a highway. When he is
looking west, the angle of depression to Exit 81 is 11 °. The exit
numbers on this highway represent the number of kilometers left
before the highway ends. What is the angle of depression, to the
nearest degree, to Exit 74 in the east?
Did you make your sketch?
I labeled exit 81 West as A and exit 74 East as B
so AB = 7 km
label the position of the balloon as P and the point below as Q
PQ = .875 km
In triangle PAQ .875/AQ = tan 11
AQ = .875/tan11 = ....
so QB = 7 - AQ
tanØ = 875/QB = ....
then find Ø
in triangle PBQ
To find the angle of depression to Exit 74 in the east, we need to first find the distance between Mike's hot-air balloon and Exit 74.
Given:
- The hot-air balloon is 875.0 m directly above the highway.
- The angle of depression to Exit 81 is 11°.
To find the distance to Exit 81, we can use trigonometry. The tangent function relates the angle of depression to the opposite and adjacent sides of the right triangle formed.
tan(angle of depression) = opposite / adjacent
tan(11°) = opposite / 875.0 m
opposite = 875.0 m * tan(11°)
opposite ≈ 174.19 m
This means the distance between Mike's hot-air balloon and Exit 81 is approximately 174.19 meters.
Now, we need to find the distance from Exit 81 to Exit 74. As per the information provided, the exit numbers on this highway represent the number of kilometers left before the highway ends.
To find the distance between Exit 81 and Exit 74, we subtract the exit numbers:
Distance = 81 km - 74 km
Distance = 7 km
Therefore, the distance between Exit 81 and Exit 74 is 7 kilometers.
Finally, to find the angle of depression to Exit 74, we can use trigonometry again:
tan(angle of depression) = opposite / adjacent
tan(angle of depression) = 174.19 m / (7 km * 1000 m/km)
tan(angle of depression) = 174.19 m / 7000 m
angle of depression = arctan(174.19 m / 7000 m)
angle of depression ≈ 1.47° (to the nearest degree)
So, the angle of depression to Exit 74 in the east is approximately 1.47°.
To find the angle of depression to Exit 74 in the east, we need to first understand the scenario.
The angle of depression is the angle formed between a horizontal line and the line of sight when looking downwards. In this case, Mike is inside a hot-air balloon and looking west, and he sees Exit 81 at an angle of depression of 11°. This means that the line of sight from Mike to Exit 81 is 11° below the horizontal line.
Now, we can use this information to determine the angle of depression to Exit 74 in the east. Since Exit 74 is in the east, the line of sight to Exit 74 will be in the opposite direction. Therefore, the angle of depression to Exit 74 will be the same as the angle of depression to Exit 81, but measured from the opposite side.
To calculate the angle of depression to Exit 74, we can subtract the angle of depression to Exit 81 from 180°. This is because the sum of the angle of depression and its corresponding angle of elevation (measured from the opposite side) is always 180°.
So, the angle of depression to Exit 74 in the east can be calculated as follows:
Angle of depression to Exit 74 = 180° - Angle of depression to Exit 81
Now we just need to substitute the values:
Angle of depression to Exit 74 = 180° - 11° = 169°
Therefore, the angle of depression to Exit 74 in the east is approximately 169°.
distance between the exits is ... d1 = 81 km - 74 km
road distance from balloon to x-81 ... d2 = 875.0 m / tan(11º)
road distance to x-74 ... d3 = d1 - d2
angle of depression to x-74 ... tan(Θ) = 875.0 m / d3