# Thread: Related Rates word problem

1. ## Related Rates word problem

I think it's a related rates problem. Anyway, I'm totally stumped on this problem and any help would be appreciated.

A man, who is 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. He 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.

Here's a (horrible) picture I drew:http://i81.photobucket.com/albums/j2...key/lolwut.jpg

2. Let's say your start point is $S$, according to your picture, we have $SP = \sqrt{(6-x)^2+2^2}=\sqrt{x^2-12x+40}$.

So your time spend on the river is $\frac{\sqrt{x^2-12x+40}}{3}$ and your time spend on land is $\frac{x}{5}$

Hence total time (in hours) can be expressed as $T(x)=\frac{\sqrt{x^2-12x+40}}{3}+\frac{x}{5}$

From here I guess you can answer the question like: determine the distance $x$ such that you can reach the house in minimum amount of time.

3. 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: . $MA = 2,\;A = 6,\;x =PB$
. . Hence: . $AP = 6-x$

From the right triangle $MAP:\;\;MP \:=\:\sqrt{(6-x)^2 + 2^2}\:=\:\sqrt{x^2-12x + 40}$
He rows that distance at 3 mph.
. . This takes him: . $\frac{\sqrt{x^2 - 12x + 40}}{3}$ hours.

Then he walks $x$ miles at 5 mph.
. . This takes him: . $\frac{x}{5}$ hours.

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

Edit: Too fast for me, Roy!

4. Well that was pretty simple. Thanks a bunch!