A line AB, 125 feet long, is measured along the sraight bank of a river. A point C is on the opposite bank. Angles ABC and BAC are found to be 65° 40' and 54° 30' respectively. How wide is the river?

Question

A line AB, 125 feet long, is measured along the sraight bank of a river. A point C is on the opposite bank. Angles ABC and BAC are found to be 65° 40' and 54° 30' respectively. How wide is the river?

Answer 1

If you drop an altitude from C to AB, call it P, and if AP=x then BP = 125-x. Then the river's width is CP. Call that w. We then have

w cotA + w cotB = 125
w = 125/(cotA+cotB)

Now just plug in your values for A and B.

Answer 2

To find the width of the river, we can use the sine rule (also known as the law of sines). The sine rule 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, let's denote the width of the river as "x". We can set up the following equation using the sine rule:

sin(BAC) / AB = sin(ABC) / AC

Let's convert the given angles from degrees to decimal form:

Angle BAC = 54° 30' = 54 + (30/60) = 54.5 degrees
Angle ABC = 65° 40' = 65 + (40/60) = 65.67 degrees

Now we can substitute the values into the equation:

sin(54.5) / 125 = sin(65.67) / x

To solve for x, we rearrange the equation:

x = (125 * sin(65.67)) / sin(54.5)

Using a calculator, we can evaluate the expression on the right side:

x ≈ (125 * 0.9190) / 0.8201

x ≈ 139.59 feet

Therefore, the width of the river is approximately 139.59 feet.

Answer 3

To find the width of the river, we can use the concept of trigonometry and apply the Law of Sines. 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 width of the river as "x."

In triangle ABC, we can use the Law of Sines:
a/sin(A) = b/sin(B) = c/sin(C)

Here,
a = BC (the width of the river, which is x)
A = angle BAC = 54° 30' = 54.5°
b = AC (the given length of line AB) = 125 feet
B = angle ABC = 65° 40' = 65.67°

Using the Law of Sines, we have:
x/sin(54.5°) = 125/sin(65.67°)

Now, let's solve for x.

x/sin(54.5°) = 125/sin(65.67°)

Cross-multiplying, we get:
x * sin(65.67°) = 125 * sin(54.5°)

Dividing both sides by sin(65.67°), we find:
x = (125 * sin(54.5°)) / sin(65.67°)

Using a scientific calculator, we can find that:
sin(54.5°) ≈ 0.8222
sin(65.67°) ≈ 0.9136

Substituting these values back into the equation, we get:
x ≈ (125 * 0.8222) / 0.9136

Calculating this expression, we find:
x ≈ 112.416

Therefore, the width of the river is approximately 112.416 feet.