Question

Two people stand on opposite ends of a bridge that crosses a river. Both can see an island located in the river. The angle

formed at the location of the first person is 37' and the angle formed at the second person is 55'. If the bridge is 5280 feet
long, how far is the first person from the island?

Answer 1

The first person is about 4327.76 feet away from the island.

Explanation:

We can draw a diagram to represent the situation. This is attached below.

Essentially, we want to find the value of x.

We are given that ∠A is 37° and ∠B is 55°.

The interior angles of a triangle must total 180°. Hence:

`m\angle A+m\angle B+m\angle C=180`

Substitute:

`(37)+(55)+m\angle C=180`

Thus:

`\displaystyle m\angle C=88^\circ`

We are also given that the length of the bridge is 5,280 feet. Since we want to find x, we can use the Law of Sines:

`\displaystyle (\sin(A))/(a)=(\sin(B))/(b)=(\sin(C))/(c)`

The variables do not really matter. It is more important that we align the angles with their respective (i.e. opposite) sides.

The respective side to ∠C is AB which measures 5280.

The respective angle to x is ∠B, which is 55°.

Hence:

`\displaystyle (\sin(55))/(x)=(\sin(88))/(5280)`

Solve for x. Cross-multiply:

`x\sin(88)=5280\sin(55)`

Thus:

`\displaystyle x=(5280\sin(55))/(\sin(88))`

Use a calculator:

`\displaystyle x\approx 4327.76\text{ feet}`

The first person is about 4327.76 feet away from the island.