MAC 2233
Section 1.4a
Functions and Linear Models
1. Linear Cost Function: C(x) = mx + b
C(x) is the total cost of producing x items. The fixed cost is b. The marginal cost is m— the cost of xth item.
2. Linear Revenue Function: R(x) = mx
R(x) is the total revenue produced by selling x items at a price of m dollars each. The marginal revenue is m.
m is the selling price of one item; the amount the total revenue increases with the sell of each additional item.
3. Linear Profit Function: Profit = Revenue – Cost
A positive profit indicates a gain; negative profit indicates a loss. When the profit is 0, when Revenue = Cost,
we say that we “break even”
4. Break-Even Point: The number of items x at which break even occurs. To determine the break-even
point, let
Profit = 0 or Revenue = Cost and solve resulting equation. (See blue box, page 43.)
EXAMPLES:
Type 1: Fixed cost and marginal cost (cost for one item) are stated.
1. A snow cone machine rents for $25 per day. The cost to make one snow cone is $.10. The cones will be
sold for $.50 each. a. Give the Cost Function. b. Give the Revenue Function. c. Profit Function.
2. a. Find the cost of producing 100 snow cones. b. What is the cost of the 100th snow cone.
3. Given the following cost and revenue functions: (a) What are the fixed costs? (b) What is the marginal
cost? (c) What is the cost of producing the 50th item? (c) Give the Profit function. (d) Find the break-even
point.
C(x) =85x + 3500; R(x) = 135x
A furniture manufacturer has a daily fixed cost of $2000 and a marginal cost of $800 per piece of furniture.
Each piece of furniture is sold for $1800.
4. a. Give the cost function. b. On a given day, what is the cost of manufacturing 4 pieces of furniture?
5. a. What is the cost of manufacturing the fourth piece of furniture? b. The twentieth piece of furniture?
6. a. Give the revenue function. b. Give the profit function. c. What is profit when 8 pieces are sold?
7. You are setting up a business with monthly costs of $50,000. Each item can be produced for $6.00. The
selling price of the item is $35. How many pieces of furniture must be sold to break even?
8. Your college newspaper has fixed costs of $100 per edition plus additional costs of 30 cents per paper. The
paper sells for 50 cents per copy. Find the cost, revenue, and profit functions. What profit or loss results from
selling 400 papers? How many copies must be sold in order to break even?
9. Interpret: (a) The marginal cost is $10. (b) The marginal revenue is $35. (c) If the fixed costs are $1000,
write the cost function, revenue function and profit function. (d) How many items must be sold to break even?
Type 2: Fixed cost and marginal cost not stated. You must use point-slope formula to write linear
model. To write a linear model, use the point-slope formula: y - y1 = m(x – x1). Time is usually x-value.
10. Roy can produce 100 CD players for $2000. He can produce 150 CD players for 3000. He sells each CD
player for $35. Give (a) cost function. (b) revenue function (c) profit function (d) break-even point.
11. Study Example 2, page 43—including the graphs.
12. ACME manufactures 150 rockers for $10,500. ACME manufactures 200 rockers for $13,000. He sells the
rockers for $75 each. Give (a) cost function (b) revenue function (c) profit function (d) Determine the cost to
manufacture 75 rockers. (e). Determine the cost of manufacturing the 75th rocker. (f) What is the profit when
1000 rockers are sold?
THE LINEAR DEMAND FUNCTION:
q(p) = mp + b
q is the number (quantity) of items demanded—the quantity that people will buy—at the price p per item.
(a) As p, the price of one item, goes up, what would you expect to happen to q, the quantity that people will
buy? (b) So if the demand for T-shirts, measured in daily sales, is given by q(p) = -4p + 90 where p is the sale
price of one shirt, interpret the slope, that is when the price of a shirt increases by $1, what happens to the
demand? (c) If the T-shirts were given away, what would be the demand?
NOTE THAT m, SLOPE, IS UNITS OF Y PER UNIT OF X.
13. Ro’s Grocery can sell 400 cans of Spicy brand baked beans for $.50 per can, but can only sell 350 cans for
$75 per can. (a) Determine the linear demand function using the point-slope formula. (b) Interpret the slope.
(c) If Ro’s gives the cans away, what would the demand be?
14. In 1994 $0.8 billion was spent on U.S. made Global Positioning System products. In 2000, $8.3 billion was
spent on these products. (a) Use this information to model total sales of GPS-based products as a linear
function of time t measured since 1994 (for 1994, t = 0; for 2000, t = ?). (b) Interpret the slope. (c) Use the
model to predict when sales of GPS-based products will reach $13.3 billion.
Homework: Finish this Worksheet plus Page 54: Odds 1 – 6 Evens 2 – 6, 15-18 all
(054)