Linear Cost, Revenue, Profit, Supply, and Demand Complete the following questions to investigate different types of linear models. Record your responses on this worksheet and the answer sheet. Turn in one answer sheet per team. Be sure to answer all parts of every question in this packet, even if there is not a spot for it on the answer sheet. You may keep this packet for studying. Part I The cost to manufacture a sofa is $600 per sofa plus a fixed setup cost of $4,500. Each sofa sells for $750. 1. What is the cost to manufacture 20 sofas? Hint: Remember to include the setup cost along with the manufacturing cost for 20 sofas at $600 each? 2. What is the cost to manufacture x sofas? 3. How much revenue is generated from selling 20 sofas at $750 each? 4. How much revenue is generated from selling x sofas? 5. How much profit does the manufacturer gain (or lose) by manufacturing and selling 20 sofas? Checkpoint: Did you find that the company loses $1500? If not, subtract the answer to 1 from 3. 6. How much does the manufacturer gain (or lose) by manufacturing and selling x sofas? C(x) = cx + F, the Total Cost function, gives the total cost for manufacturing x units at a unit cost of c and fixed costs F. (The money paid out by the company.) R(x) = sx, the Revenue function, gives the total revenue realized from manufacturing and selling x units at the selling price s. (The money brought in by the company.) P(x) = R(x) C(x) = sx (cx + F) = (s c)x F, the Profit function, gives the total profit realized from manufacturing and selling x units. (The net amount of money the company will have after paying all of its expenses.) Texas A&M University Page 1
The linear Cost, Revenue, and Profit functions for this problem are: C(x) = 600x + 4500 R(x) = 750x P(x) = 150x 4500 Hint: These are the same functions you should have found in 2, 4 and 6. 7. How many sofas must be sold in order to have a profit of $12,000? 8. The Cost, Revenue, and Profit functions are graphed below on the same grid. Determine which graph corresponds to each function. C(x): R(x): P(x): 9. Which of these three linear business models should always contain the origin? Why? Texas A&M University Page 2
Part II Demand The quantity demanded of a computer monitor is 7,500 units when the unit price is $750. At a unit price of $700, the quantity demanded increases to 9,000 units. Assume this relationship is linear. Let x be an independent variable representing the number of monitors consumers are willing to buy (the quantity demanded). Let p be the dependent variable representing the unit price. Express ordered pairs as (x, p). 1. If the unit price is $750, what is the value of x? Write the corresponding ordered pair. 2. If the quantity demanded is 9,000 units, what is the value of p? Write the corresponding ordered pair. 3. Find the slope of the line determined by the two points in 1 and 2. Checkpoint: Did you find m = - 1? If not, remember that slope is found by dividing the change in the y 30 values (in this case, p values) by the change in the x values. 4. Find the equation of the line determined by the two points found in 1 and 2. Hint: The point-slope formula for the equation of a line is y y 1 = m(x x 1 ), where m is the slope of the line and (x 1, y 1 ) is one of the points on the line. Use p instead of y in your work. The line you just found is a demand equation. A demand equation expresses the relationship between the unit selling price and the quantity demanded by consumers. 5. Consider the demand equation from 4, a. Should the values of x ever be negative? Why or why not? b. Should the values of p ever be negative? Why or why not? c. Consider the graph of the demand function. From left to right, is the line falling or rising? Is the slope positive or negative? The characteristics you just described always hold for demand equations. Variable values must make sense, and since consumers will buy less of a product when the price is higher, slope is negative. Texas A&M University Page 3
Supply The manufacturer will not market any of the computer monitors if the price is $600 or lower. However, for each $50 increase in the unit price above $600, the manufacturer will produce 1000 additional units. Assume this relationship is linear. Let x be an independent variable representing the number of monitors suppliers are willing to produce. Let p be the dependent variable representing the unit price. Express ordered pairs as (x, p). 6. How many monitors is the manufacturer willing to produce at a price of $600? In other words, if the unit price is $600, what is the value of x? Write the corresponding ordered pair. 7. What is the value of x at a price of $650? Write the corresponding ordered pair. Checkpoint: (x, p) = (1000, 650) 8. What is the value of x at a price of $700? Write the corresponding ordered pair. 9. Find the slope of the line determined by the points you found in 6 and 7. 10. Find the equation of the line determined by the points found in 6 and 7. Use slope-intercept form to express your answer. Hint: The y-intercept (in this case p-intercept) is given in 6. The line you just found is a supply equation. The supply equation expresses the relationship between the unit selling price and the quantity supplied by the producers. 11. Consider the supply equation from 10, a. Should the values of x ever be negative? Why or why not? b. Should the values of p ever be negative? Why or why not? c. Consider the graph of the supply function. From left to right, is the line falling or rising? Is the slope positive or negative? The characteristics you just described always hold for supply equations. Variable values must make sense, and since manufacturers will make more of a product when the price is higher, slope is positive. Texas A&M University Page 4
Part III At a unit price of 33 dollars, 5600 picture frames will be demanded and 6900 picture frames will be supplied. If the unit price is decreased by $12, consumers will purchase an additional 1600 picture frames, while manufacturers will provide 3600 fewer picture frames. Assume these relationships are linear. 1. Find two points on the graph of the linear demand function. 2. Find two points on the graph of the linear supply function. 3. Write the linear equations that could be used to represent supply and demand. Texas A&M University Page 5