optimization problem

Question

Your car runs out of gas on a road running north/south at a point 3 miles from the highway (which runs east/west). The nearest gas station is 6 miles east down the highway from the point your road meets the highway. You can walk down the road(s) at 4 miles per hour, or straight across the fields at 3 miles per hour.

It is Sunday at 1:48pm, and the gas station closes at 4pm. What is your fastest walking route to the gas station? Can you make it there before the gas station closes?

Answer 1

Raj,

 

A couple of initial thoughts to get you started:

 

1. Go ahead and draw the picture if you didn't already.  There's a north-south road, 3 miles long, and then an east-west road, 6 miles long.  You're standing at the south end of the north-south road, and you're trying to get to the gas station at the far end of the east-west road.
2. The idea is that you don't necessarily have to walk diagonally along the field, or just along the roads.  You can walk a little bit north, and then cut diagonally across the field.
3. To show what that looks like, draw a short segment north along the road, and then connect that point diagonally to the gas station.

 

The question now is:  how long does it take you to walk along that route?  We don't know how far north you want to walk, so let's call that "x." 

 

1. For the first leg of the trip, you're walking along the road, so you're walking at 4 mph.  With distance equal to rate times time, you can get your time to walk "x" miles at 4 mph.
2. The second leg of the trip is a diagonal.  If that's the hypotenuse of a right triangle, and you can write expressions for the two legs, then you can figure out that distance in terms of x.  The east-west leg has a known distance, given in the problem.  The north-south leg of the triangle isn't the full 3 miles...because you've already walked x miles of it.  It's a right triangle, so the pythagorean theorem should help from there.
3. Once you have the diagonal distance, and a rate of 3mph trough the fields, you can get an expression for the time for that part of the trip.
4. Your total time is the expression for those two legs, put together...
5. ...and you're trying to minimize the time.  In calculus, how do you find local minima?

 

Even though you won't make it to the gas station in time by going straight through the field or following the roads the whole way, I bet you can get there by finding a happy medium between the two, and that's what your time equation will allow you to do.

 

Let us know if that's enough to get you going...hope this helps!

 

 

Answer 2

The distance you travel along the roads is 3 miles + 6 miles = 9 miles.  The distance across the field is √(3²+6²) = 3√5 ≅ 6.7 miles.

 

distance = rate*time

distance/rate = time

 

For the road route:

 

(9 miles)/(4 mph) = 2 hours 15 minutes

 

For the field route:

                                 

(6.7 miles)/(3 mph) = 2 hours 14 minutes

 

So cutting across the field is one minute faster than walking along the roads.  You have 4 pm - 1:48 pm = 2 hours 12 minutes to get to the gas station before until it closes.  Looks like you'll have to call a cab no matter which route you take.