# Math Help - One last optimization problem! Island problem

1. ## One last optimization problem! Island problem

A small island is 3 miles from the nearest point P on the straight shoreline of a large lake. If a woman on the island can row a boat 2 miles per hour and can walk 3 miles per hour, where should the boat be landed in order to arrive at a town 12 miles down the shore from P in the least time? Let be the distance between point P and where the boat lands on the lakeshore. Hint: time is distance divided by speed.

(A) Enter a function that describes the total amount of time the trip takes as a function of the distance .
= (sqrt(x^2+9))/2+(12-x)/3

(B) What is the distance that minimizes the travel time? Note: The answer to this problem requires that you enter the correct units.
= .

(C) What is the least travel time? Note: The answer to this problem requires that you enter the correct units.
The least travel time is .

(D) Recall that the second derivative test says that if and , then has a local minimum at . What is ?
=

I got the first one, which is (sqrt(x^2+9))/2+(12-x)/3

also how do I derive that? do I used chain rule for the first function?

2. You can simplify this to:

$T(c) = \dfrac12 (x^2 + 9)^{1/2} + \dfrac{12}{3} - \dfrac13 x$

You can derive each separately now?

3. I got the derivitive so now I set it to zero? What do I do for the other problems? I am very confused...

4. Originally Posted by softballchick
I got the derivitive so now I set it to zero? What do I do for the other problems? I am very confused...

set the derivative = 0 , find the critical value of x and use the 2nd derivative test to determine if the critical value is a max or min.

also ... please do not bump. if someone is interested in helping, they will.