Johnny Steamboat wants to sail from his island home to town in order to purchase a book of carpet samples (for reasons unknown). His home island is 7 miles from the nearest point on the shore. The town is 35 miles downshore and one mile inland (shore is represented by the horizontal line in the diagram). If he can run his steamboat at 12 mph and catch a cab as soon as he reaches the coast that will drive at 60 mph (obviously some kind of weird 4x4 cab that drives in a straight line, despite the existence of roads)  ...

(a)    How far down the coast is the point that Johnny should hit land to minimize the time it takes to get to town?

(b)    What do you think Mr. Steamboat is going to do with those carpet samples anyhow?

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Solution:

(a)    Consider the figure two right triangles. The left-hand portion of the horizontal line we will call x, and therefore the right-hand portion of the line is 35x. The actual distance traveled is given by the diagonal line. We'll call the upper portion of the diagonal D and the lower (shore) portion d. We want to minimize time, so we need a function which gives time traveled. As you should know, d = rt. Thus, t = d/r. Using the Pythagorean Theorem to get D and d, we set up this time function:

(b)    The carpet samples are needed to make a hairpiece for the theatrical revival of Joseph and the Amazing Technicolor Toupee.

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