Read the following carefully and complete the tasks. Organize or summarize your answers in a word document. Use complete sentences and label units when appropriate. Be sure to include your first and last name, student ID and section number in the header of your paper. There is not a length requirement for this assignment, but you want to make sure that all of the information is included in your report and make sure that the information is accurate. Print and submit your assignment to the instructor by the end of the class period on Friday, October 4. You can submit the assignment to the instructor before or after class or to the instructor’s mailbox (located near Neill 103, open M-F 8am-5pm). Late assignments will not be accepted. Please staple multiple pages together. (staple only, no paperclips, etc.)

Suppose you start your own business manufacturing and selling a product or service. Suppose that you are running this business out of your home, so you will consider the fixed costs to include your current rent and utilities. If you do not pay rent/utilities, please interview someone to determine a reasonable fixed cost for this region. You can round your fixed cost up to the nearest whole dollar amount. Then come up with an appropriate and reasonable product or service and variable cost per unit to manufacture your product or service. Next, you’ll need to come up with two pairs of data and use these to write the price-demand equation. Decide on two different prices which you could reasonably sell your product to consumers. For each price, decide how many items you would reasonably make/sell within one month. Your goal with these two data points is to create a price-demand equation in the form P=MX + B to describe the relationship between the demand of your product/service, X, and the consumer price, P, in dollars. Recall that your price-demand equation should be a decreasing linear function. Write your function such that all coefficients including M and B are fractions or integer values, instead of using the decimal approximation.

Now, write a linear cost function to describe the total monthly cost for your business. Use your price-demand equation to write the revenue function to describe your business’ total monthly revenue for selling X products/services. Then find the total profit function. Recall that the cost, revenue and profit functions should be written in terms of X and simplified completely. Again, be sure the coefficients are in exact form using fractions or integer values and avoid using the decimal approximation for the coefficients of each of the functions. Determine the lowest possible production level and the highest possible production level.

You must verify that your functions are compatible by graphing the cost and revenue functions together on the same plot. You can use the graphing utility, desmos.com/calculator, to graph these functions. You can edit the graph options by selecting the wrench icon in the upper right corner. Adjust the values for the X-axis and Y- axis so that quadrant I is in view of the screen and you can easily read the graph. You can use the lowest and highest production levels that you just found to adjust the X-axis. Adjust the Y-axis to include the peak of the revenue function in your graph. Verify that your revenue and cost functions intersect in two points within quadrant I. If this does not happen, go back and adjust your values or data points so that the revenue and cost function intersect in two points. Take a screenshot of the graph and provide it in your report. Find the break-even points, rounding to two decimal places. Be as precise as possible by using exact values throughout your calculations and rounding only at the end. How many products should you produce in order to see profit for your business and production levels will cause your company to incur a loss?

 Be specific by using lower and upper bounds in your answers. If your product or service is something that should be produced and sold as a whole unit, be sure to consider this when you are defining these production levels. Reflect back to the two pairs of data that you used to write your price-demand equation. Do either of the demand values lie in the region where your company will produce a profit? If so, select one to represent your current monthly production level. If neither of your two points of data falls into the range of units where your company will profit, then choose a production level that is. Then determine the total cost, revenue and profit at this production level. Find the instantaneous rate of change of cost, revenue and profit at this production level and write a sentence or two to interpret each of these values. Round your answer to two decimals. Using the marginal profit, should you increase, decrease or keep the production level the same in order to achieve maximum total profit? Use differentials to approximate the total profit if production is increased by one unit. Then calculate the exact total profit of producing one more unit. Discuss how close your approximate value is to the exact value you found. Find the production level where the tangent line to the profit function is horizontal. What is the importance of this value? Make a conclusion about what production level you should choose in order to optimize your company’s total monthly profit.