In order to build a bridge across a river, a surveyor is hired to find the distance across the river. The surveyor places a stake on either side of the river. As shown in the diagram below (the diagram is not drawn to scale). He measures the bearing to the first stake as 16 degrees East of North at a distance of 562 feet. He measures the departure of the second stake as 330.87 feet West and the latitude as 198.80 feet North. Find the distance across the river. Hint: label the diagram and use the information given to help form a triangle then apply one of the laws to solve for the missing side

Question

In order to build a bridge across a river, a surveyor is hired to find the distance across the river. The surveyor places a stake on either side of the river. As shown in the diagram below (the diagram is not drawn to scale). He measures the bearing to the first stake as 16 degrees East of North at a distance of 562 feet. He measures the departure of the second stake as 330.87 feet West and the latitude as 198.80 feet North. Find the distance across the river. Hint: label the diagram and use the information given to help form a triangle then apply one of the laws to solve for the missing side

Answer 1

Interpretation of given information:

The location of the theodolite (surveyor) at point A.
First stake (Point B)
Second stake (Point C).
Points B and C are on opposite sides of the river, and distance BC represents the width of the river.
For point B:
distance = 562
bearing = 16° (from north clockwise)
For point C:
departure (Easting) = 330.87'
Latitude (northing) 198.8'
equivalent to:
distance = √(330.87²+198.8²)
=386.0005
bearing=90-atan(198.8/330.87)=61.6135°

Let's form triangle ABC,
AB=562'
AC=386.0005'
∠BAC=61.6135-16=45.6135°

Use cosine rule to find distance BC
BC²=562²+386.0005²-2(562)(386.0005)cos(45.6135°)
=161354.304
BC=√(161354.304)=401.7'

Width of bridge = 401.7'

(Do check the calculations for accuracy because I used the Google calculator with which I am not familiar.)

Answer 2

To find the distance across the river, we can use the given information and apply the law of sines to solve for the missing side of the triangle.

Let's label the diagram as follows:
- The first stake is labeled as A.
- The second stake is labeled as B.
- The location of the bridge across the river is labeled as C.

Now let's break down the given information:

- The bearing to the first stake from the surveyor's position is 16 degrees East of North.
- The distance to the first stake (AB) is 562 feet.
- The departure of the second stake from the first stake is 330.87 feet West (horizontal distance from A to B).
- The latitude of the second stake from the first stake is 198.80 feet North (vertical distance from A to B).

We can represent the distance across the river as side BC.

Now let's use the law of sines to solve for BC:

First, we need to find angle BAC. We have the bearing, which is 16 degrees East of North. Since this angle is measured clockwise from North, it is 90 degrees - 16 degrees = 74 degrees.

Now, let's use triangle ABC and the law of sines:

sin(A)/AC = sin(B)/BC

We already have the value of angle BAC, which is 74 degrees. So we can rewrite and substitute in the values:

sin(74 degrees)/562 = sin(90 degrees)/BC

To solve for BC, we can cross multiply and divide:

BC = (562 * sin(90 degrees)) / sin(74 degrees)

Using a scientific calculator, we can calculate:

BC ≈ (562 * 1) / 0.9613

BC ≈ 584.2 feet

Therefore, the distance across the river (BC) is approximately 584.2 feet.