A boat is sailing due east parallel 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. Find the distance from the boat to the shoreline if the lighthouse is at the shoreline.

Question

I did 70 + 63 = 133
then 10sin(133)=7.31

My teacher said this is wrong but I have no idea what to do.

Answer 1

Let's start with a new diagram, since I have a feeling yours is all wrong.
(there is no logical reason to add 70 and 63)

Draw two parallel lines, the upper one is the path of the boat, the bottom one is the shoreline.
Pick a point A, the first position of the boat, and then a bit to the right, pick a point B, the second position of the boat.

You can draw dotted perpendiculars from A and B to the bottom line to create 90° angles.
Pick point L, for lighthouse, on the bottom line to the right of both A and B.
So we are now looking at triangle ABL
angle LAB = 20° , (took 90 - 70)
and angle LBA = 153 (took 90 +63)
since the 3 angles must add up to 180 , angle ALB = 7°
and finally, AB = 2.5 miles (10miles/hour x 1/4 hour)

so now by the Sine Law,
BL/sin20 = 2.5/sin7
BL = 7.016 (I will let you confirm that)

Almost done.....
From L draw a dotted line to the other parallel line to meet it at C
LC is the distance we want!
triangle BCL is right=angled with angle CBL = 27°
sin 27 = LC/BL

LC = BLsin27
= 7.016(sin27)
= 3.185

Answer 2

Thank You