Chapter A, Problem 60E

Athletics An athlete has set up a course in which she is dropped off by a boat 2 miles from the nearest point on shore. Once she reaches the shore, she must run to a point 4 miles down the coast and 2 miles inland (see figure). She can swim 2 miles per hour and run 6 miles per hour. The time $t$ (in hours) required for her to complete the course can be approximated by the model

$t=\frac{\sqrt{x^2+4}}{2}+\frac{\sqrt{{\left(4-x\right)}^2+4}}{6}$

where $x$ is the distance (in miles) down the coast from her starting point to the point at which she leaves the water to start her run.

(a) Use a table to approximate the distance down the coast that will yield the minimum amount of time required for the athlete to complete the course.

(b) The expression below was obtained using calculus. It can be used to find the minimum amount of time required for the triathlete to reach the finish line. Simplify the expression.

$\frac{1}{2}x{\left(x^2+4\right)}^{-1/2}+\frac{1}{6}\left(x-4\right){\left(x^2-8x+20\right)}^{-1/2}$