Hello, bob9171!

This is a classic problem, found in every Calculus text.

A man, in a boat 2 miles from the nearest point A on a straight shoreline,

wants to reach a house located at point B which is 6 miles further down the coast.

He plans to row to point P that is between point A and B.

P is also x miles from the house.

He will then walk the remainder of the distance to the house.

Suppose he can row at a rate of 3 mph and can walk at a rate of 5 mph.

If T is the total time to reach the house, express T as a function of x. Code:

M *
| *
| *
2 | *
| *
| *
* - - - - - * - - - - - - *
A 6-x P x B

We are told that: .$\displaystyle MA = 2,\;A = 6,\;x =PB$

. . Hence: .$\displaystyle AP = 6-x$

From the right triangle $\displaystyle MAP:\;\;MP \:=\:\sqrt{(6-x)^2 + 2^2}\:=\:\sqrt{x^2-12x + 40}$

He rows that distance at 3 mph.

. . This takes him: .$\displaystyle \frac{\sqrt{x^2 - 12x + 40}}{3}$ hours.

Then he walks $\displaystyle x$ miles at 5 mph.

. . This takes him: .$\displaystyle \frac{x}{5}$ hours.

His total time is: .$\displaystyle T \;=\;\frac{\sqrt{x^2-12x+40}}{3} + \frac{x}{5}$ . hours.

Edit: Too fast for me, Roy!