a surveyor wishes to measure the distance between two objects, a carabao and a tower on the opposite side of the river. while standing at point R, he finds that the distance between the carabao is 30 meters, while the distance to the tower is 415 meters. The angle formed between them is 13 degrees and 25 minutes. how far is the carabao to the tower?
i am dealing with the applications of oblique triangles. Please help me by answering my questions. thank you very much and God bless!:)
This is a very common problem for surveyors to find distances between two objects which they cannot reach. The same principle is used to find difference in heights.
Surveyors can do this today because the newer generation of theodolites can measure distance without a prism placed at the distant object.
Here we have a triangle CRT (C=carabao, R=surveyor, T=tower) in which ∠CRT=13°25', CR=30m, RT=415m.
Since two sides and the included angle are all known, the cosine rule is useful.
If the unknown side is r, RT=c (side opposite to angle C, CR=t (side opposite to angle T, then the cosine rule gives
r²=c²+t²-2ct(cos(R)).
Calculate r² and hence r using your calculator.
To solve this problem, we can use the Law of Sines to find the length of the side opposite the given angle.
Using the Law of Sines formula:
sin(A) / a = sin(B) / b = sin(C) / c
Where A, B, and C are the angles and a, b, and c are the sides opposite those angles.
In this case, we know angle C (the angle between the carabao and the tower) is 13 degrees and 25 minutes. First, we need to convert this angle to decimal form:
13 degrees + (25 minutes / 60 minutes/degree) = 13.42 degrees.
Next, we can plug in the values into the Law of Sines formula:
sin(A) / 30 = sin(13.42) / c
Now, we can solve for c by isolating it:
c = (30 * sin(13.42)) / sin(A)
Next, we need to find angle A. To do that, we can use the fact that the sum of the angles in a triangle is 180 degrees:
A + C + 90 degrees (since the distance is measured from a right angle) = 180 degrees
A = 180 degrees - C - 90 degrees
Now, we can calculate the value of A:
A = 180 degrees - 13.42 degrees - 90 degrees
A = 76.58 degrees
Now, we can plug in the values to calculate c:
c = (30 * sin(13.42)) / sin(76.58)
Using a scientific calculator, we find that sin(13.42) is approximately 0.2311 and sin(76.58) is approximately 0.9785.
c = (30 * 0.2311) / 0.9785
c ≈ 7.096
Therefore, the distance between the carabao and the tower is approximately 7.096 meters.
To solve this problem, we can use the law of sines to find the distance between the carabao and the tower. The law of sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant.
Let's label the distance between the carabao and the tower as "c", the distance between the surveyor and the carabao as "a", and the distance between the surveyor and the tower as "b". The given information is as follows:
a = 30 meters (distance between the surveyor and the carabao)
b = 415 meters (distance between the surveyor and the tower)
angle A = 13 degrees 25 minutes (angle between the carabao and the tower)
First, we need to convert the angle from degrees and minutes to decimal degrees. To do this, we can use the following formula:
Decimal degrees = degrees + (minutes / 60)
In this case, the angle A in decimal degrees would be:
13 + (25 / 60) = 13.4167 degrees
Now, we can use the law of sines to find the length of side c. The law of sines states:
c / sin(C) = a / sin(A)
Where C is the angle opposite side c. In this case, angle C is the angle between the surveyor and the carabao, which is 180 degrees - (A + B). Angle B is the angle between the surveyor and the tower, which can be found using the following formula:
B = 180 - A - C
Thus, we can calculate angle B as follows:
B = 180 - A - C
B = 180 - 13.4167 - 180
B = -13.4167
Since the sum of the angles in a triangle is always 180 degrees, this negative angle implies that the triangle is not possible. It means that there is an error in either the given measurements or the calculations.
Please double-check the given measurements and try solving the problem again.