Distance From Boat to Shoreline

[nycmathguy]

Section 6.1

Looking for the set up only.

 

[MathLover1]

Since the boat is traveling at a rate of 10 miles per hour (60 minutes), that same boat travels 2.5 miles in 15 minutes.

Draw a diagram. [On the diagram shown, all angles are in degrees.] This diagram should show two triangles -- one with a 72° angle to the lighthouse, and another with a 66° angle to the lighthouse.

Find the complementary angles of 18° and 24°.
The angle immediately under the boat's present location measures 66°+90°=156°.
For the angle with the smallest measure in the diagram, I have used the fact that
6°=24°−18°, but you may also subtract the sum of 156° and 18° from 180°.

This gives us an oblique triangle whose angles measure 156°,18°,and 6° and one of whose sides measures 2.5 miles.

You may now use the Law of Sines to find the direct distance to the lighthouse.

sin6°/2.5=sin18°/d

This gives a direct distance of approximately 7.4 miles.
If you want the perpendicular distance to the shore, you may now use basic trigonometry. If y is the perpendicular distance, then

y/7.4=sin23°

y=7.4/sin23°

This is approximately 2.9 miles.

 

[nycmathguy]

Since the boat is traveling at a rate of 10 miles per hour (60 minutes), that same boat travels 2.5 miles in 15 minutes.

Draw a diagram. [On the diagram shown, all angles are in degrees.] This diagram should show two triangles -- one with a 72° angle to the lighthouse, and another with a 66° angle to the lighthouse.

Find the complementary angles of 18° and 24°.
The angle immediately under the boat's present location measures 66°+90°=156°.
For the angle with the smallest measure in the diagram, I have used the fact that
6°=24°−18°, but you may also subtract the sum of 156° and 18° from 180°.

This gives us an oblique triangle whose angles measure 156°,18°,and 6° and one of whose sides measures 2.5 miles.

You may now use the Law of Sines to find the direct distance to the lighthouse.

sin6°/2.5=sin18°/d

This gives a direct distance of approximately 7.4 miles.
If you want the perpendicular distance to the shore, you may now use basic trigonometry. If y is the perpendicular distance, then

y/7.4=sin23°

y=7.4/sin23°

This is approximately 2.9 miles.

Very nice. Thank you.