Correcting a Navigational Error

[greatwhiteshark]

A yacht leaves St. Thomas bound for an island in the British West Indies, 200 miles away. Keeping a constant speed of 18mph but encountering heavy crosswinds and strong currents, the crew finds that after 4 hours that the sailboat is offcourse by 15 degrees.

a) How far is the boat from the island at this time?

I found this distance to be 131.8 miles using law of cosines. To find 131.8 miles, I first multiplied 18mph by 4 hours to find distance AB on a triangle that I drew. I multiplied 18 x 4 = 72 miles (distance from A to B).

b) Through what angle should the sailboat turn to correct its course?

My answer is: 20.15 degrees but my teacher said the answer is 23.1 degrees. How is this possible?

c) How much time has been added to the trip because of this? (Assume that the speed remains at 18 mph.)

I did not get part c at all. I recall Soroban helping with a similar question but HONESTLY, part c is a problem.

 

[tkhunny]

b) Through what angle should the sailboat turn to correct its course?

My answer is: 20.15 degrees but my teacher said the answer is 23.1 degrees. How is this possible?

Sadly, you did not describe your method, so I cannot tell you where 20.15 comes from.

Law of Sines

131.7776/sin(15º) = 200/sin(b) ==> b = 23.1296º
This is a little misleading, as the triangle clearly has an obtuse angle opposite the 200 mi side. Remember that the arcsin function is often defined with a Range of (-pi/2, pi/2). It will NOT give you an obtuse angle, so one must also remember that sin(aº) = sin(180º-aº). However, if we switch to the supplementary angle to solve the triangle (156.8704º), and then calculate the corresponding external angle, the required course correction, we get back to where we started, 23.1296º.

Part C) The extra time is very simple. You'll kick yourself...

Distance = Rate * Time

Intended

200 mi = 18 mph * Time₁

Realized

(72 mi + 131.7776 mi) = 18 mph * Time₂

The answer will be in hours, Time₂ - Time₁. Multiply by '60 min/hr' to get minutes.

Go get'em.

 

[greatwhiteshark]

Yes...

You are right about wanting to kick myself. Sometimes I miss the obvious trying to rush through the question.