A boat is sighted from of two lighthouses which are one mile

A boat is sighted from of two lighthouses which are one mile along a straight east-west shoreline. From the first lighthouse, the angle between the boat and due east is 55 degree. From the second lighthouse, the angle between the boat and due east is 70 degree. How far is the boat from the shore? Exact answer plus units. This is a variant of the problem in Lecture 25. Let d be the distance between the boat and the nearest point on the shoreline. Let x be the distance between the closest lighthouse and the nearest point on the shore line. There are two right triangles in the picture. Apply tangent to the two given angles to get two equations involving x and d. Solve these two equations for d.

Solution

The distance of 1st lighthouse = 1+x

distnace of 2nd lighthouse = x

Applying tan rule in bigger triangle::

tan55 = d/(x+1) ----(1)

1.43 = d/(x+1)

Applying tan rule in smaller triangle::

tan70 = d/x ----- (2)

2.74 = d/x

Solvd the equations:

1.43(x+1) = d

2.74x = d

-------(subtract)

1.43x +1.43 -2.74x =0

1.31x = 1.43

x = 1.091 miles

d = 2.74x = 2.99 = 3.0 miles