At 7AM on a windless summer day, The SS Sophie Germain set out on its maiden voy

Question

At 7AM on a windless summer day, The SS Sophie Germain set out on its maiden voyage. The captain of the ship set his bearings for 25 degrees West of North (N25°W). The vessel sailed at this heading at a rate of 20 miles per hour, docking at 11:45 AM. The captain and his crew had a leisurely lunch at the Cartesian Café, who’s slogan “I drink, therefore I am” rang true for the weary travelers. At 2 pm, the Sophie Germain resumed its seaward journey, heading 12 degrees East of North (N12°E) at a rate of 25 miles per hour. The Germain docked safely at 5:30 pm. The captain and his crew got a room for the night at the nearby Newton Manor.

1. Assuming no wind on the following day, what heading should The Germain follow to return directly to its original port?

2. How many miles from the original port, is the SS Sophie Germain when it sets out on its return voyage?

3. When the Germain began its homeward journey the following day, the captain neglected to account for the wind which was blowing 12 miles per hour 35 degrees East of North (N35°E). The captain pointed his vessel in the direction of his original port and sailed at 25 miles per hour for one hour and 45 minutes before realizing his mistake. a. How far from his original port was The SS Sophie Germain when the captain realized his mistake? b. Assuming the wind remains the same, what direction should the captain point his ship now, to head directly to port?

4. The captain fixes his heading according to the answer to number (3b) and returns home. a. How many miles would The Germain have traveled from the time it left port on that windless summer morning to the time it returned home, if the captain had correctly accounted for the wind on the homeward trip? b. How many extra miles did The Germain travel as a result of the captains error?

5. Draw a model of the situation. Include vectors representing each leg of the trip. Label each vector. (Drawing must be to scale). Use a blue pen to mark the route the captain should have taken (without the wind miscalculation) and a red pen to mark the route the captain did take.

I need help with number 3 and beyond. I have the triangle with one side being 173 and the magnitude of the wind plus the boats movement is 31.56. How do I find the angle between these sides?

Explanation / Answer

1) In first stint he travels 95 miles if we take two components : we get 42.3 miles west and 86.1 miles north in 2nd stint he travels 87.5 miles if we take 2 components we get 18.2 miles east and 85.6 miles north add these two we get 14.1 miles west and 171.7 miles north direction to take to get to the port : 4.7 degrees east of south total miles from original port = sqrt(14.1^2 + 171.7^2) = 172.3 miles.

2) 7.287 degrees east of south

3)effective velocity after adding velocity of wind : x -direction = -15cos7.287 + 10 cos25 = -5.815 miles/hr y direection = -15sin7.287 - 10sin25 = -6.128 miles/hr final coordinate after 45 minutes = (167.325,17.36)

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