Solve set 45 has arrived! This week’s Solve is all about Sea Exploration. Challenge yourselves to think creatively as your sharpen your skills related to lines on a sphere, trig functions, angles of incidence and more!
Ease yourself into the set with Question 1: Are We There Yet?
Christopher Columbus’ first voyage to the Americas was all made possible by a series of miscalculations. In 1492 after seven years of rejections, the Queen of Spain agreed to his expedition with an estimated travel distance of 2,400 miles (3,900 km). Previously Portugal and other kingdoms had previously rejected his proposal because they calculated the distance to be far longer. Actually, over 1,500 years earlier, Erastothenes had calculated the radius of the Earth to be about 4,500 miles (7,000 kilometers). It was also commonly accepted that the Europe-Africa-Asia landmass spanned about 180 degrees of latitude. On October 12, 1492, after departing from the Canary Islands, Christopher Columbus sighted land which he mistook for an island off the coast of China, but in reality was what we know as The Bahamas. Columbus sailed straight west from the Canary Islands, which are at 28° North latitude. How many times his estimate would he have had to sail if he really needed to go halfway around the world?
Up for a Challenge? Try Question 5: But the Wind is Goin’ the Rudder Direction:
Ships of the 15th century were powered by sails, which function at the mercy of natural wind direction and speed. Suppose that your destination is directly East, but the wind is blowing directly from the East. It is possible to reach your destination, even though the wind is blowing directly from it. Your ship’s keel and rudder can force your ship to travel only forward or backward along a fixed direction, so by choosing that direction carefully, and setting the correct angle for your sails, it is possible to zig-zag back and forth towards your destination.
Assume for simplicity that your sail is a perfectly flat plane, and the force imparted on your ship is entirely determined by Newton’s Third Law from wind perfectly reflecting off your sail, and that this results in a thrust vector equal to the projection of that force onto the direction parallel to your keel. If you wish to maximize the projection of that thrust vector in the direction of your destination (due East), what is the optimal angle you should choose to set your keel at, measured in degrees of deviation from due East?