. A surveyor wishes to measure the distance across Pasig River. She sets up her transit at a point C on the bank of the river, and sights on a point B on the other bank directly opposite her. Then she turns the transit through a right angle, and measures off a distance of 100 meters to a point A. She moves the transit to A, and measures the angle CAB, which is 50deg 11min 10 sec. How wide is the river?

Question

. A surveyor wishes to measure the distance across Pasig River. She sets up her transit at a point C on the bank of the river, and sights on a point B on the other bank directly opposite her. Then she turns the transit through a right angle, and measures off a distance of 100 meters to a point A. She moves the transit to A, and measures the angle CAB, which is 50deg 11min 10 sec. How wide is the river?

Answer 1

To find the width of the river, we can use trigonometry and the given information.

1. First, draw a diagram to visualize the scenario:

A
|
|
| 50° 11' 10"
| /\
|/ \
B_____\_____________
C X

C: Surveyor's position on the bank of the river
A: Transit position after measuring 100 meters
B: Point on the opposite bank of the river
X: The point directly below transit position A

2. Use the information given to find angle BAC:
Since the transit is turned through a right angle (90 degrees), angle BAC is the complementary angle to angle CAB.
Angle BAC = 90° - 50° 11' 10"

3. Convert the angle BAC to decimal degrees:
1 degree = 60 minutes = 3600 seconds
The angle in decimal degrees = 90 + 11/60 + 10/(60 * 60)

4. Use trigonometry (specifically the tangent function) to find the width of the river:
tan(angle BAC) = (width of the river) / (distance from A to B)
width of the river = (distance from A to B) * tan(angle BAC)

5. Plug in the values:
Given: distance from A to B = 100 meters
angle BAC in decimal degrees (as calculated in step 3)

6. Calculate the width of the river:
width of the river = 100 * tan(angle BAC)

Now, perform the calculations to find the width of the river using the given information.

Answer 2

To find the width of the river, we can use trigonometry and the given information. Let's break down the steps:

Step 1: Set up a diagram
Draw a diagram with the points A, B, and C:

A ------ B
|
|
|
C

Step 2: Label the known values
Based on the given information, label the known values on the diagram:
- The distance from C to A is 100 meters (AC = 100m).
- The angle CAB is 50 degrees 11 minutes 10 seconds (CAB = 50° 11' 10").

Step 3: Determine the angle CBA
Since we have a right-angled triangle (CAB), we can determine the angle CBA by subtracting CAB from 90 degrees:

CBA = 90° - CAB
= 90° - 50° 11' 10"

To perform the subtraction, convert the angle CAB from degrees, minutes, and seconds to decimal form:
50° 11' 10" = 50 + (11/60) + (10/3600) = 50.1861 degrees (rounded to four decimal places).

Now, subtract CAB from 90° to find CBA:
CBA = 90° - 50.1861°
= 39.8139°

Step 4: Use trigonometry to find the width of the river
In triangle ABC, we have the length of the side AC (100m) and the angle CBA (39.8139°). We want to find the length of side AB, which represents the width of the river.

Using the trigonometric function tangent (tan), we can write the following equation:

tan(CBA) = AB / AC

Rearranging the equation:

AB = AC * tan(CBA)

Plugging in the values:
AB = 100m * tan(39.8139°)
AB ≈ 100m * 0.8264

Calculating AB, we get:
AB ≈ 82.64 meters

Therefore, the width of the river is approximately 82.64 meters.

Answer 3

draw the diagram, and it is clear that the distance x is

x/100 = tan 50°11'10"