the pont du gard near Nimes,france, is a roman aqueduct.an observer in a hot air balloon some distance away from the aqueduct determines that the angle of depression to each end is 54 degrees and 71 degrees,respectively.the horizontal distance between the balloon and the closest end of the aqueduct is 270.0m. calculate the length of the aqueduct to the nearest tenth of a metre
No, trig :)
Draw a sketch, h is height of balloon over bridge
tan 71 h/270
so
h = 270 tan 71 = 784
then
tan 54 = h/x where x is hor dist to other end from balloon
x = h/tan 54 = 570
so total length = 270 + 570 = 840 meters
To calculate the length of the aqueduct, we can use trigonometry and apply the tangent function.
Let's consider the observer in the hot air balloon. The observer measures the angle of depression to each end of the aqueduct, which tells us the angle between the observer's line of sight and the horizontal ground.
First, let's label the given information:
- Angle of depression to the closest end of the aqueduct: 54 degrees
- Angle of depression to the farthest end of the aqueduct: 71 degrees
- Horizontal distance between the balloon and the closest end of the aqueduct: 270.0 m
Next, we can determine the vertical distances from the balloon to each end of the aqueduct using the tangent function.
For the closest end of the aqueduct:
tan(54 degrees) = vertical distance / 270.0 m
Rearranging the equation, we have:
vertical distance = 270.0 m * tan(54 degrees)
Using a scientific calculator, calculate the value of tan(54 degrees). It gives you approximately 1.3763819.
So, the vertical distance to the closest end of the aqueduct is:
vertical distance = 270.0 m * 1.3763819 = 371.009 m (approximately)
For the farthest end of the aqueduct, we follow the same steps:
tan(71 degrees) = vertical distance / 270.0 m
Rearranging the equation, we have:
vertical distance = 270.0 m * tan(71 degrees)
Using a calculator, calculate the value of tan(71 degrees). It gives you approximately 2.7374862.
So, the vertical distance to the farthest end of the aqueduct is:
vertical distance = 270.0 m * 2.7374862 = 738.596 m (approximately)
Now that we have the vertical distances, we can calculate the length of the aqueduct by subtracting the vertical distance to the closest end from the vertical distance to the farthest end:
Length of the aqueduct = vertical distance to the farthest end - vertical distance to the closest end
= 738.596 m - 371.009 m
= 367.587 m (approximately)
Therefore, the length of the aqueduct is approximately 367.6 meters to the nearest tenth of a meter.