| A group of farmers want to take their grain to one of two local elevators. One elevator is at point E
(0,0) and the other is at point F (30,0), where each unit represents 1 mile. The elevator at point E
will pay a farmer $pE per bushel and the elevator at point F will pay a farmer $pF per bushel.
Assume that any farmer can transport his or her grain from the farm to either elevator for $0.02 per
bushel per mile. The profit per bushel is then given as: profit = p - cd, where p is the price paid per
bushel by the elevator, c is the cost of transportation per bushel per mile and d is the distance
traveled to the elevator. Maximizing the profit per bushel is equivalent to maximizing the total profit. Each farmer, of course, wants to maximize profit. Look at a particular farmer who lives d1 miles from point E and d2 miles from point F. This farmer wants to know when it will be more profitable to bring grain to point F than point E. 1. Assume that each elevator will pay $3.10 per bushel for grain on Monday. Find the relationship between d1 and d2 based on this. You should start by setting up an equation that relates the 2 profits available. 2. Describe geometrically and algebraically all points where the farmer could live so that, in the case described in number 1, the profit will be equal no matter which elevator buys the grain. 3. Assume that the elevator at point E will pay $3.50 per bushel for grain on Tuesday, but the elevator at point F will still only pay $3.10 per bushel. Find a relationship between the difference in the distances d1 and d2 based on this. Proceed as you did in number 1, but this time, solve for (d1 - d2). 4. Recall the definition of a hyperbola. If points E and F are considered the foci and the difference of the distances from the farmer to each point is known (from number 3), the equation of a hyperbola can be written. Write this equation. Graph the hyperbola. 5. What do all of the points on the hyperbola in number 4 represent in terms of the farmers mentioned and their profits? 6. Describe geometrically where a farmer should live, with respect to the graph in number 4 so that the profit obtained from taking the grain to the elevator at point F is actually more than the profit obtained from taking it to point E. |