Problem $7$ Suppose you are going on a weekend trip to a city that is d miles away. Develop a model that determines your gasoline costs for the round trip. What assumptions or approximations are necessary to treat this model as a deterministic model? Do you find these assumptions or approximations acceptable?

Problem $8$ Recall the production model from Section $1\mathrm{.}3$: Suppose the firm in this example considers a second product that has a unit profit of $$5$ and requires $2$ hours for each unit produced. Suppose total production capacity is still $40$ units. Use and as the number of units of product $2$ produced.

a$.$ Show the mathematical model when both products are considered simultaneously.

b$.$ Identify the controllable and uncontrollable inputs for this model.

c$.$ Draw the input$-$output process flowchart for this model $($see Figure $1\mathrm{.}5).$

d$.$ What are the optimal solution values $$of x and y$?$

e$.$ Is this model a deterministic or stochastic model? Explain Problem $13$ Micromedia offers computer training seminars on a variety of topics. In the seminars, each student works with a personal computer and practices the specific activity that the instructor is presenting. Micromedia is currently planning a two$-$day seminar on using Microsoft Excel in statistical analysis. The projected cost for the seminar is $$600$ per student. The cost of the lecture hall, instructor compensation, lab assistants, and promotion is $$9,600.$

Micromedia rents computers for its seminars at a cost of $$60$ per computer per day.

a$.$ Develop a model for the total cost of hosting the seminar. Let x be the number of students who enroll in the seminar.

b$.$ Develop a model for the total profit if x students enroll in the seminar.

c$.$ Micromedia has forecast an enrollment of $30$ students for the seminar. How much profit will be made if your forecast is accurate?

d$.$ Calculate the break$-$even point.

$$ Hand in Case Problem Scheduling a Golf League Chris Lane, the head professional at Royal Oak Country Club, must develop a schedule of matches for the couples golf league that begins its season at $4$:$00$ p$.$m$.$ tomorrow. Eighteen couples have signed up for the league, and each couple must play every other couple over the course of the $17-$week season. Chris thought it would be easy enough to develop a schedule, but after working on it for a couple of hours, he has been unable to come up with one. Since Chris must have a schedule ready by tomorrow afternoon, he has asked you to help him. One possible complication is that one of the couples has told Chris that they may have to cancel the season. They told Chris that they would let him know by $1$:$00$ p$.$m$.$ tomorrow whether they will be able to play this season.

Manager's Report Prepare a report for Chris Lane. Your report must include, at a minimum, the following items: $1.$ A schedule that allows for each of the $18$ couples to play against all other couples during the $17-$week season.

$2.$ A contingency schedule that can be used if the couple that contacted Chris decides to cancel the season.