On page 1, there is data you must fill in before proceeding. The information listed below  is the data 

F value =950 , V value =8.4, p1 value =16  ,x1 value =224 ,p2 value =23   , x2 value =112
You need to download this file, and then modify it by entering your initial data and then typing in your answers to all the questions.
You might need to generate a graph or two (we recommend desmos.com) and import them.

1

Math202 Spring 2020 Project

Section Number:

Last name, First Name

Read the following carefully and complete the tasks. 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 summarized 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, March 13.

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. 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. After consulting a financial advisor, you find out the following:
(NOTE: Values for the quantities below are given in Blackboard, under MyGrades.)

· Your fixed cost is F =

dollars per month.

· Your variable cost is V = dollars per product.

· If the product is priced at p1 = dollars, you can make and sell x1 = units in a month.

· If the product is priced at p2 = dollars, you can make and sell x2 = units in a month.

Now you can find your linear cost function C(x) using the given costs, and create a price-demand equation using the two data points () and from above. Your price-demand equation should be in the form to describe the relationship between the demand for your product, , and the consumer unit price, , in dollars. Recall that your price-demand equation should be a decreasing linear function. Make sure to use fractions or integer values, instead of using the decimal approximation. Then find the marginal cost function.

Cost Function:

Marginal Cost Function:

Price-Demand Equation:

Your work / explanation for the above answers:

Now, use your price-demand equation to write the revenue function to describe your business’ total monthly revenue from selling units. Then find the total profit function, marginal revenue function and marginal profit function. The revenue and profit functions should be written in terms of and simplified completely. Again, be sure the coefficients are in exact form using fractions or integer values, so avoid using the decimal approximation for any of the coefficients.

Revenue Function:

Profit Function:

Marginal Revenue Function:

Marginal Profit Function:

Your work / explanation

for the above answers:

Determine what the highest possible production level is, according to your price-demand equation. Also 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. Determine the range of production levels that will result in positive profit for your business. Also determine the corresponding unit price range that will result in positive profit.

Highest possible production level:

Break-even points:

For positive profit: < x < , which corresponds to < p <

Your work / explanation for the above answers:

Reflect back to the two pairs of data that you used to write your price-demand equation. Choose one of the two demand values. (Let a = the demand value you choose.) Now suppose your current monthly production level is a units. Then determine the total cost, revenue and profit at this production level and find the marginal cost, revenue and profit at this production level a. Write a few sentences to interpret each of these values. Round your answer to two decimal places.

At production level a = :

Total Cost is . Marginal Cost is .

Total Revenue is: . Marginal Revenue is .

Total Profit is : . Marginal Profit is .

Your work / explanation

Consider the marginal profit at production level a you found in the previous step. Determine whether you should increase, decrease or keep the production level the same in order to achieve maximum total profit. Use differentials to approximate the total profit from producing and selling (a+1) units. Then calculate the exact total profit from producing and selling (a+1) units.

To achieve maximum total profit, the production level should be: (Select one below)

Increased /decreased / kept the same .

If the production level a is increased by 1,

The approximate total profit is:

The exact total profit is:

Your work / explanation

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.

Production level you should choose:

Your work / explanation for the above answer:

Using an online graphing utility such as Desmos.com/calculator, graph the functions C(x) and R(x) on the same coordinate plane. Plot and label the coordinates of the intersections. Explain what those intersections represent. On a separate x-y coordinate plane, graph the function P(x). Plot and label the coordinates of the vertex. Explain what the vertex represents.

Make sure your graphs are large and clear, so that the points indicated above are easily read. Include both of these graphs in this report (see note below).

Your work / explanation / answers for the above:

Note: At desmos.com, a graph can be downloaded as follows: (1) click the little boxed arrow to the upper-right of your graph, (2) hit “Export image”, (3) hit “Download PNG”, (4) save the file (it may go to your Downloads folder by default). Then, in your Word document, put the cursor where you want the image to be, select the “Insert” tab, then select “Image”, then select the saved image file (it’s a file) from the appropriate location on your computer (possibly your Downloads folder). **You can erase this note before printing your project**.