Individual Project
Suppose you have a start-up company that develops and sells a gaming app for smartphones. You get to know the financial performance of this by understanding your cost, revenue, and profit.
The monthly cost function (in US dollars) of developing your app is C(x) = 3x + b, where C(x) is the cost and x is the number of app downloads. The $3 in the equation is called the variable cost per unit, and b is called the fixed cost.
The monthly revenue function (in US dollars) based on previous monthly sales is modeled by the function R(x) = -0.15x2 + 153x, 0 ≤ x ≤ 500.
The monthly profit function (in US dollars), P(x), is derived by subtracting the cost from the revenue in the function P(x) = R(x) - C(x).
For each question, be sure to show all your work details for full credit. When applicable, round all value answers to the nearest cent.
1. Based on the first letter of your last name, choose a value for your fixed cost, b. What is your cost function, C(x)?
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First letter of your last name
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Possible values for b
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A-F
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$1,000-$1,500
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G-L
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$1,501-$2,000
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M-R
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$2,001-$2,500
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S-Z
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$2,501-$3,000
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2. If P(x) = R(x) - C(x), what is your profit function?
3. Using the vertex formula, at how many downloads would your company achieve a maximum profit? How much is this maximum profit?
4. How many downloads would give you a profit of $10,000 the next month? (Hint: Let the profit function be equal to $10,000 and solve for x using the quadratic formula.)
5. Complete the table below by calculating the profit for each number of downloads. (Show your work.)
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x, number of downloads
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P(x), profit in US dollars
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0
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100
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200
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300
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500
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6. Using the table from part 5, generate a graph of your profit function using Excel or another graphing utility. Insert the graph into the supplied Student Answer Form. Be sure to label and number the axes appropriately so that the graph matches the chosen and calculated values from above.
7. Why do you think it is important to understand your start-up company's financial performance?
References
Desmos. (n.d.). Retrieved from https://www.desmos.com/
Graph 4.4.2. (n.d.). Retrieved from the Graph Web site: http://www.padowan.dk/
Mathematics 4.0. (n.d.). Retrieved from the Microsoft Web site: https://www.microsoft.com/en-us/default.aspx