Suppose you work as a dispatcher for Marten Transport, a local trucking company. Part of your job is to determine the recommended driving (constant) speed in order to minimize the total operating cost for the company. The following information about operating costs has been collected:

1. A truck travels over a 100 mile portion of flat interstate.

2. When the truck travels at a constant rate of 50 miles per hour gets an (average) fuel mileage of five miles per gallon.  For each mile per hour increase in the speed, the truck's mileage decreases by one-tenth mile per gallon.

3. Diesel fuel costs $1.08 per gallon.

4. The fixed cost for use of the truck is $12 per hour.

5. Marten Transport pays the truck drivers $30 per hour in wages.

Submission:

A graph of total operating cost as a function of speed on the interval (50 mph ,70 mph) and the constant speed between 50 mph and 70 mph (if such a value exists) that drivers should maintain on the 100 miles of straight interstate for the lowest total operating cost. Using the language of calculus, explain why your answer gives the most economical total operating cost.

Submission worksheet:

In a beehive, each cell is a regular hexagonal prism, open at one end with a trihedral angle at the other end. It is believed that bees form their cells in such a way as to minimize the surface area for a given volume, thus using the least amount of wax in cell construction, Examination of these cells has shown that the measure of the apex angle theta is amazingly consistent. Based on the geometry of the cell it can be shown that the surface area is given by  where s , the length of the sides of the hexagon, and h , the height, are constant.

Submission a):

1. Calculate .

2. What angle should be bees prefer?

3. Determine the minimum surface area of the cell (in terms of s and h ).

4. Pick , and and plot the function . Choose other values of s and h and plot the function.

Submission worksheet:

The manager of a 100-unit apartment complex knows from experience that all units will be occupied if the rent is $400 per month. A market survey suggests that, on the average, one additional unit will remain vacant for each $5 increase in rent.

1. Calculate the rent that their manager should charge to maximize revenue.

2. Include a good graph of the revenue as a function of the rent.

Submission worksheet:

3. Application to optimization.

Two towns are located near a river, one town is 10 miles upstream from the other town. The upstream town is two miles from the river and the downstream town is 5 miles from the river. The two towns agree to co-operate in building a pumping station on the river to supply water to both towns. The two towns wish to locate the pumping stating at a site on the river that would minimize the amount of pipeline necessary to deliver water to both towns. We can use Maple to draw a sketch of this situation.

> with(plottools):

Warning, the name arrow has been redefined

> with(plots): 

Warning, the name arrow has been redefined

> p1:=polygon([[0,0], [10,0], [10,5],[6,0],[0,2]], linestyle=1, thickness=2): 

> p2:=textplot([[0,2.3,`A`],[10,5.3,`B`],[6,.3,`P`],[-.5,1.0,`2 mi.`],[10.6,2.5,`5 mi.`]],color=blue):

> p3:=textplot([[3,-0.5,`X`],[8,-0.5,`10-X`]],color=black):

> display(p1,p2,p3,axes=none);

If you let the horizontal distance from the pump house at point P to the point on the river nearest point A by x , then the horizontal distance to the point on the river nearest point B is equal to . The total length of the pipeline is

> L:=x->sqrt(x^2+4)+sqrt((10-x)^2+25);

To optimize this, we are interested in finding where L reaches a minimum on the domain [0,10]. We first consider where .

> solve(D(L)(x)=0,x);

> evalf(20/7);

Notice that the derivative is defined for all x , so that the minimum occurs either at this point or one of the end points, and it is easy to determine what the minimum value is and where it occurs.

> evalf(L(0));evalf(L(20/7));evalf(L(10));

From the values above, we determine that the minimum length of pipeline required is about miles, and we should position the pump house at about 2.85 miles away from the point on the river nearest point A.

Submission:

Two towers are 150 feet apart, one tower is 50 feet high and the other is 30 feet high. Two guy wires are run from a point on the ground between the two towers, one to the top of each tower.

(a) Copy the procedure above to draw a picture of this situation.

(b) Find where the point between the two towers should be located to minimize the total length of the guy wires.

(c) Show that in general that regardless of the height of the towers, the length of the guy wires is minimized if the angles formed at the point on the ground are equal.

Submission worksheet: