Answer
(a) $T(x)=\frac{\sqrt {4+(7-x)^2}}{2}+\frac{x}{5}$
(b minimum $T=2.317$ hours when $x=6.127$ miles.
Work Step by Step
(a) For the right triangle with AP as hypotenuse, one side is the river width $2mi$, another side is $7-x$ as indicated in the figure of the problem. So $\bar {AP}=\sqrt {2^2+(7-x)^2}$ and the total time (T) needed is given by
$T(x)=\frac{\bar {AP}}{2}+\frac{x}{5}=\frac{\sqrt {4+(7-x)^2}}{2}+\frac{x}{5}$
(b Graph the above function as shown in the figure. A minimum in time can be found as $T=2.317$ hours when $x=6.127$ miles.
