Question

Need help with setting up an equation for this problem:

A boat is sailing due east parellel to the shoreline at a speed of 10 miles per hour.At a given time the bearing to the lighthouse is S 70 degrees E, and 15 minutes later the bearing is S 63 degrees E. The lighthouse is located at the shoreline. Find the distance from the boat to the shoreline.

Answer 1

I drew 2 parallel lines, one for the shoreline, the other for the path of the boat.

I labeled two points on the boat-line as A and B, and made AB = 2.5 miles (10*15/60)

I labeled a point L on the shoreline for the lighthouse, so that angle BAL = 20º and angle ABL = 153º
which makes angle ALB = 7º.

so by the Sine Law
AL/sin153º = 2.5/sin7º
AL = 9.313

Now draw a perpendicular from A to the shoreline at C.
Then triangle ACL is rightangled, with AL=9.313 and angle A = 70º.

so cos70=AC/9.313
AC = 3.185

So the boat is 3.185 miles from shore