1.      In solving this problem, you should state the objective function and the constraint equation(s), and then determine the best route:

 

A farmer owns a very long field that is 200 meters wide.  There are roads that run along the field’s edges but none that cut through it.  The farmer needs to walk from one side of the field to a point on the other side and 400 meters down the field.  His speed is 5 km/hr when walking on a road, but only 3km/hr when walking through the muddy field.  Determine the route he should take to get to his destination as quickly as possible, and how long this will take him.

 

The farmer first walks to some point on the far side of the field and then walks the rest of the way (if any) to his desitnation along the road on the far side.  (Note:  We could just as easily say he walks along the road on the near side of the field for some distance first, and then aims to his destination point – it will be an equivalent problem).  Suppose he aims for a point C on the far side, and C is x meters down the field from his starting point (in addition to being on the far side).  Then he walks 200 m=0.2km across the field, and x km down the field, for a total distance of  in the mud.  This takes . 

He then completes his journey by walking 0.4-x km along the road, which takes him

So his total time, as a function of x, is

This is our objective function that we need to minimize.

Setting this equal to 0, we get

(we ignore -.15 as it is not in our meaningful domain)

So the only critical point is when he aims .15km =150 meters down the field.

T(.15) =

 

Comparing this to the endpoints:

If x=0 (he walks straight across), he walks 0.2 km in the field, and 0.4 km on the road, for a time of

If x=.4 (he walks straight toward B), he doesn’t walk on the road at all, and his distance in the field is

 

 

2.      A minor league baseball team has discovered that when it charges $10 per ticket, its average attendance is 2000, but when it charges $9 per ticket, its average attendance is 2500. 

a.       Find demand function for these tickets, assuming that it is linear.

b.      Determine the price that the team should charge, and the number of tickets that it will sell, in order to maximize its revenue.