An observer in a hot air balloon some distance away from the aqueduct determines that the angle of each depression to each end is 54 degrees and 71 degrees. The closest end of the aqueduct is 270 m from the balloon. Calculate the length of the aqueduct to the nearest tenth of a metre
Answer: 299.8 m
I confused, i tried using sine law
these were my calculations but some how i'm not getting the right answer
sin54/270=sin 109/x
1st we draw a picture of the prob.
1. Draw a ver and hor line to form a rt
angle.
2. Draw the hyp. from top of ver line to the end of hor line.
The angle between the hyp and hor line = 54 deg.
3. Draw a 2nd hyp from the top of ver
line to a point on the hor line.The
distance from this point to bottom of
the ver line is 270m. The angle formed
is 71 deg. The dist. between the 2 points where the 2 hyp cross the hor
line is X.
tan54 = h / (270+X),
h = (270 + X)tan54,
tan71 = h / 270,
h = 270tan71,
h = (270 + X)tan54 = 270tan71,
Divide both sides by tan54:
270 + X = 270tan71 / tan54 = 569.71,
X = 569.71 - 270 = 299.7m.
To solve this problem using the sine law, you need to use the correct angle. The first angle you provided, 54 degrees, should be opposite the side of the aqueduct that you are trying to calculate. The second angle, 71 degrees, should be opposite the known side, which is the distance from the balloon to the closest end of the aqueduct (270 m).
Let's call the length of the aqueduct "x".
Using the sine law, the correct calculations would be:
sin(54°) / 270 = sin(71°) / x
Rearranging the equation, we get:
sin(54°) = (sin(71°) * 270) / x
Now, let's solve for x:
x = (sin(71°) * 270) / sin(54°)
Calculating this expression, we get:
x ≈ (0.9455 * 270) / 0.8090
x ≈ 252.135 / 0.8090
x ≈ 311.698
Therefore, the length of the aqueduct is approximately 311.7 meters, rounded to the nearest tenth of a meter.
Note: The value you provided as the answer, 299.8 m, seems to be a rounded value. However, using the given information, the more accurate answer is approximately 311.7 m.
To solve this problem, you can use the trigonometric relationship known as the Sine Law. However, there seems to be a mistake in your calculation. Let's go through the correct steps together:
First, label the lengths and angles we have:
Length of the side of the aqueduct nearest to the balloon (opposite the 54-degree angle) = x
Length of the side of the aqueduct farthest from the balloon (opposite the 71-degree angle) = 270 m
Using the Sine Law, we can write the following equation:
sin(54 degrees)/270 = sin(71 degrees)/x
Now, cross multiply to get:
x * sin(54 degrees) = 270 * sin(71 degrees)
To solve for x, divide both sides of the equation by sin(54 degrees):
x = (270 * sin(71 degrees))/sin(54 degrees)
Using a scientific calculator or an online calculator, calculate the value of the right side:
x ≈ (270 * 0.9397) / 0.8090
x ≈ 312.70 / 0.8090
x ≈ 386.19 m
Therefore, the length of the aqueduct to the nearest tenth of a meter is 386.2 m, not 299.8 m as mentioned in your answer. It appears there might have been a mistake in the given solution.