The Pont du Gard near Nîmes, 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° and 71°, respectively. The horizontal distance between the balloon and the aqueduct is 270.0 m. Calculate the length of the aqueduct to the nearest tenth of a metre.
Use sine law?
Sine law? Oh, you mean the ruler of all the sines! Well, since you brought it up, let's dive into the crazy world of math and solve this problem using the sine law!
First, let's label the sides and angles of this crazy aqueduct triangle. The length of the aqueduct, which we're trying to find, will be side 'c'. The angles of depression will be angle 'A' (54 degrees) and angle 'B' (71 degrees). The distance between the balloon and the aqueduct will be side 'a' (270 meters).
According to the sine law, we have:
sin A / a = sin B / c
Plugging in the values we know:
sin 54° / 270 = sin 71° / c
Now, let's do some magical math to solve for 'c' by cross-multiplying:
sin 54° * c = sin 71° * 270
Dividing c on one side of the equation:
c = (sin 71° * 270) / sin 54°
Bust out your calculator and let's crunch some numbers...
c ≈ 306.9 meters
So, to the nearest tenth of a metre, the length of the aqueduct is approximately 306.9 meters. I hope that brings you as much joy as a clown juggling calculus balls!
Yes, we can use the sine law to solve this problem.
The sine law states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. In this case, we can use the sine law to relate the angles of depression to the length of the aqueduct.
Let's call the length of the aqueduct "x" (in meters). From the given information, we can establish the following diagram:
A
/ |
/ |
/ | x
/ θ1 |
/ |
/_____|
B 270m C
θ2
In the diagram, A represents the location of the observer in the hot air balloon, B and C represent the ends of the aqueduct, θ1 represents the angle of depression to end B, and θ2 represents the angle of depression to end C.
According to the sine law, we have:
sin(θ1) = BC / AB and sin(θ2) = BC / AC
Rearranging, we get:
BC = AB * sin(θ1) and BC = AC * sin(θ2)
Since the horizontal distance between the balloon and the aqueduct is 270.0 m (AB), we can substitute this value into the equations:
BC = 270.0 * sin(θ1) and BC = AC * sin(θ2)
To find the length of the aqueduct (x = BC), we need to find the value of AC. We can rearrange the second equation to solve for AC:
AC = BC / sin(θ2)
Substituting the value of BC from the first equation, we get:
AC = (270.0 * sin(θ1)) / sin(θ2)
Now we can calculate the length of the aqueduct (x = BC) by substituting the value of AC and θ1 into the first equation:
x = 270.0 * sin(θ1)
Now, using the given values of θ1 = 54° and θ2 = 71°, we can calculate the length of the aqueduct (x):
AC = (270.0 * sin(54°)) / sin(71°)
x = 270.0 * sin(54°)
Calculating these values, we find:
AC ≈ 291.6 m
x ≈ 221.3 m
Therefore, the length of the aqueduct to the nearest tenth of a metre is approximately 221.3 meters.
To solve this problem, we can indeed use the Sine Law. The Sine Law relates the lengths of the sides of a triangle to the sines of its angles.
Let's label the length of the aqueduct as "x". We can then form a triangle with the aqueduct as the base, and the observer's line of sight from the balloon to each end of the aqueduct as the other two sides.
Using the given information, we have:
- Angle A = 54°
- Angle B = 71°
- Side a (opposite angle A) = x (the length of the aqueduct)
- Side b (opposite angle B) = 270.0 m (the horizontal distance between the balloon and the aqueduct)
According to the Sine Law, we have the following relationship:
a/sin(A) = b/sin(B)
Substituting the given values, we get:
x/sin(54°) = 270.0/sin(71°)
Now, we can solve for x by rearranging the equation:
x = (270.0 * sin(54°)) / sin(71°)
Using a calculator, evaluate the sine of 54° and 71°, and substitute the values into the formula.
After calculating, you should find that the length of the aqueduct (x) is approximately 406.5 m when rounded to the nearest tenth of a metre.