Jackrabbit Trails is a family camp located in Kings Canyon National Park, California. Open from mid-May to Labor Day, Jackrabbit Trails offers week-long family summer vacation packages that include lodging, three meals a day, and numerous activities such as horseback riding, waterskiing, sailing, canoeing, archery, fishing, hiking, and mountain biking. To provide families with a restful experience, Jackrabbit Trails limits the number of families to 20 per week.

Moreover, the camp motto is “Follow the fun without following the herd.” Jackrabbit Trails believes that the following function captures the “true” underlying relation between the number of families at the camp in a given week and the total costs of running the camp for a week:

Total cost ($) = 1,000 + 300 (Families) - 20 (Families)2 + (Families)3

Required:

a. Plot (graph) Jackrabbit Trails’ weekly total costs as a function of the number of families for 0 to 20 families. Does the shape of this cost function look familiar to you (think back to microeconomics)? What are some of the properties of this cost function?

b. Suppose Jackrabbit Trails approximates its cost function via a linear model using the endpoints of its relevant range from 4 families to 16 families per week. Write out the linear cost model and plot it on the same graph you developed in part (a). Over what range does the linear cost function strike you as a reasonable approximation to Jackrabbit Trails’ true underlying cost curve?

c. Suppose Jackrabbit Trails believes that the following equation captures the relation between the price (per family), and weekly demand (number of families) for their summer camp:

Price = 1,200 - (50 × Families)

What price and quantity maximize Jackrabbit Trails’ weekly profit when the total cost is nonlinear? What price and quantity maximize Jackrabbit Trails’ weekly profit when the total cost is linear? What does Jackrabbit Trails sacrifice and/or gain by adopting the linear cost model?