A line AB, 125 feet long, is measure along the straight bank of a river. A point C is on the opposite bank. Angles ABC and BAC are found to be 65 degrees 40' and 54 degrees 30' respectively. How wide is the river?
Draw altitude CD, length x (we want).
let y = AD
then 125 - y = BD
tan 65 40 = x/ (125-y)
tan 54 30 = x/y
solve for x and y, you need x, the altitude.
If the width is w, then
wcotA + wcotB = 125
w = 125/(cot65°40' + cot54°30')
To find the width of the river, we can use the concept of trigonometry and the information given about the angles and the length of line AB.
Step 1: Draw a diagram to understand the given information better.
A
* '
* ' '
* ' '
* ' '
* ' angle BAC = 54° 30'
* '
* '
* '
* ----- 125 ft ----- '
C
Step 2: Identify the right triangle formed by line AB, the river width, and the perpendicular line from point C to line AB.
Step 3: Let's call the width of the river as x. Now, we can express the trigonometric relationships in terms of the given angles and the length of line AB.
In triangle ABC,
- The angle BAC is opposite to the side BC.
- The angle ABC is opposite to the side AC.
- The angle ACB is 180° - (BAC + ABC).
So, we can write:
sin BAC = BC / AB
sin(54° 30') = x / 125 ft
sin ABC = AC / AB
sin(65° 40') = x / 125 ft
Step 4: Calculate the value of sin(54° 30') and sin(65° 40').
Using a calculator, we find:
sin(54° 30') ≈ 0.8000
sin(65° 40') ≈ 0.8763
Step 5: Substitute the calculated values into the equations to find the width of the river.
0.8000 = x / 125 ft
Cross-multiplying, we get:
x = 0.8000 * 125 ft ≈ 100 ft
Therefore, the width of the river is approximately 100 feet.
To find the width of the river, we can use the Law of Sines. The Law of Sines states that the ratio of the sine of an angle to the length of the side opposite that angle is the same for all angles in a triangle.
First, let's label the given information:
- Line AB is 125 feet long.
- Angle ABC is 65 degrees 40' (or 65.67 degrees).
- Angle BAC is 54 degrees 30' (or 54.5 degrees).
We need to find the length of side AC, which represents the width of the river.
Now, using the Law of Sines, the equation is as follows:
sin(A) / a = sin(B) / b = sin(C) / c
Here, A, B, and C are the angles of the triangle (65.67, 54.5, and 180 degrees, respectively), and a, b, and c are the corresponding sides of the triangle (125 feet, AC, and the length of the river).
We need to find the length of side AC, so we can set up the equation:
sin(A) / 125 = sin(B) / AC
To solve for AC, we need to isolate it. We can cross multiply and rearrange the equation:
AC = 125 * (sin(B) / sin(A))
Next, substitute the given values:
AC = 125 * (sin(54.5) / sin(65.67))
Using a calculator, calculate the sines of the angles and substitute them into the equation:
AC ≈ 125 * (0.833 / 0.908)
Multiply and divide:
AC ≈ 125 * 0.915
AC ≈ 114.375
Therefore, the width of the river, AC, is approximately 114.375 feet.