1. ## Optimization

It's 7:50 P.M. Your snowmobile is out of gas and you are 3 miles due south of a major highway. The nearest gas station on the highway is 6 miles east of your position; it closes at 10 P.M. You can walk 4 miles per hour on the roads, but only 3 miles per hour through snow fields. Can you make it before it closes? What is the best route to minimize your time?

2. This is another Pythagorean min.

We use d=rt to minimize the time. t=d/r.

The distance through the snow is:

$\displaystyle \sqrt{9+x^{2}}$

The distance along the roa is $\displaystyle 6-x$

You are given the rates, so the time is:

$\displaystyle \frac{\sqrt{9+x^{2}}}{3}+\frac{6-x}{4}$

That is what must be minimized. Set to 0 and solve for x.

That will be the distance from the point directly across from where the snowmobile ran out of gas to where you must walk to in oreder to minimize the time.

3. Do you mean that the derivative of the equation should be set to zero to minimize it? Otherwise I'm not sure how this would be a calculus problem...

4. Originally Posted by ebonyscythe
Do you mean that the derivative of the equation should be set to zero to minimize it? Otherwise I'm not sure how this would be a calculus problem...
Yes.