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 aqueduct is 270.0 m. Calculate the length of the aqueduct to the nearest thenth of a metre.
To help you understand what I've done,
we'll draw a picture of the problem:
1. Draw a rectangle with the longest
sides in hor. positiion and the shortest sides vertically.
2. Draw a diagonal from the upper
left vertex to the lower right vertex.
3. Draw a 2nd line from the upper left
vertex to a point near the center of
the bottom line. Label the portion to
the lt. 270 m and the portion to the rt. X meters.
4. Label the angle between the top
line of the rectangle and the diag-
onal 54 deg. This is the angle of
depression.
5. At the lower right vertex, the diag.
forms 2 angles. Label the one on the left 54 deg. also. We have formed 2 rt. triangles. The angle between the
top line of the rectangle and the shortest hyp. is the 2nd angle
of depression and measures 71 deg.
6. Label the acute angle formed by the smaller hyp. and the bottom of the
rectangle 71 deg.
Tan 71 = h / 270, h = 270 Tan 71 =
784.14 m. = Height of observer.
Tan 54 = 784.14 / (X = 270),
(X + 270) Tan 54 = 784.14
X + 270 = 784.14 / Tan 54 = 569.7
X = 569.7 - 270 = 299.7 m = Length of
Aqueduct
Nada, can you send me the answer so I
can compare with mine before posting?
Thanks so much !
Well, isn't that an aqueductic puzzle! Let's put on our "aqua-culating" hats and solve it!
Based on the given information, we have two angles of depression: 54 degrees and 71 degrees. Imagine the aqueduct as a straight line that connects the two ends. The distance between the balloon and the aqueduct is 270.0 m.
Now, to calculate the length of the aqueduct, we can use some good ol' trigonometry!
First, let's find the height difference between the balloon and the aqueduct. We can use the tangent function:
tan(54 degrees) = height difference / 270.0 m
Rearranging the equation, we get:
height difference = 270.0 m * tan(54 degrees)
Next, let's calculate the horizontal distance between the two ends of the aqueduct. We can use the same trigonometric approach:
tan(71 degrees) = horizontal distance / 270.0 m
Rearranging the equation, we get:
horizontal distance = 270.0 m * tan(71 degrees)
Now, we can use the Pythagorean theorem to find the length of the aqueduct:
length of aqueduct = sqrt((horizontal distance)^2 + (height difference)^2)
Now, it's time for some number crunching. Don't worry, I've got my "aquamatic" calculator right here!
Calculating the height difference: height difference = 270.0 m * tan(54 degrees) ≈ 306.3 m
Calculating the horizontal distance: horizontal distance = 270.0 m * tan(71 degrees) ≈ 775.2 m
Now let's find the length of the aqueduct using the Pythagorean theorem:
length of aqueduct = sqrt((775.2 m)^2 + (306.3 m)^2) ≈ 842.7 m
So, the length of the aqueduct is approximately 842.7 meters. Ta-da!
To solve this problem, we can utilize trigonometry, specifically the tangent function. The tangent of an angle of depression can be used to determine the length of the side opposite the angle (in this case, the length of the aqueduct).
Let's break down the information given:
- The angle of depression to one end of the aqueduct is 54 degrees.
- The angle of depression to the other end of the aqueduct is 71 degrees.
- The horizontal distance between the balloon and the aqueduct is 270.0 meters.
Step 1: Calculate the length of the side opposite the angle of depression of 54 degrees.
Using the tangent function, we can write:
tan(54 degrees) = length of aqueduct / 270.0 meters
Rearranging the equation, we get:
length of aqueduct = tan(54 degrees) * 270.0 meters
Using a scientific calculator, calculate tan(54 degrees). The result is approximately 1.3763819204711735.
Now we can substitute this value into the equation:
length of aqueduct = 1.3763819204711735 * 270.0 meters
length of aqueduct ≈ 371.78 meters (rounded to two decimal places)
Step 2: Calculate the length of the side opposite the angle of depression of 71 degrees.
Using the same process, we have:
tan(71 degrees) = length of aqueduct / 270.0 meters
Rearranging the equation, we get:
length of aqueduct = tan(71 degrees) * 270.0 meters
Using a scientific calculator, calculate tan(71 degrees). The result is approximately 2.746481177278854.
Now we can substitute this value into the equation:
length of aqueduct = 2.746481177278854 * 270.0 meters
length of aqueduct ≈ 741.40 meters (rounded to two decimal places)
Step 3: Calculate the total length of the aqueduct.
To find the total length of the aqueduct, we need to add the lengths calculated in Step 1 and Step 2:
Total length = 371.78 meters + 741.40 meters
Total length ≈ 1113.18 meters (rounded to two decimal places)
Therefore, the length of the aqueduct, to the nearest tenth of a meter, is approximately 1113.2 meters.