MAT 151
Applications to Economics (Marginality)—Section 4.5
I. Basic Economics TermsBasic Functions
q is the quantity of goods in units
p is the price per unit
R is the revenue from q unitsR(q)
Sketch TWO reasonable revenue functions’ graphs
C is the cost of producing q unitsC(q)
Sketch TWO reasonable cost functions’ graphs
P is the profit from selling q units P(q) =
Discuss the geometric interpretation of profit for the graphs above.
GOAL: Determine the quantity which maximizes profit.
Investigate using derivatives…
Marginal revenue (extra revenue per unit):
Marginal cost (extra cost per unit):
Marginal profit (extra profit per unit):
Restated condition necessary for max profit:
EXAMPLES on reverse side; please work one per team and submit your work to me at the end of class. I will scan and post them. You should study the work of the other teams.IIa. Angela owns a volleyball business where the fixed costs are $1000, variable costs are $2.50 per volleyball, and she sells the volleyballs for $7.95 each. Write a formula for total cost as a function of quantity q, and write a formula for revenue as a function of quantity q. How many volleyballs must she sell in order to break even?
aIII. Mike owns a small bicycle business. He finds that his marginal cost C'(50)=100 and his marginal revenue R'(50)=85. Should he increase or decrease his production of bicycles from 50 units? Why?
IVa. Julia owns a small business that makes TU t-shirts. It costs her $4 each to make the shirts, and her revenue from selling q shirts is R(q)= 50q-q2. She also has fixed costs of $200 per month. Find her profit function and help her decide how many t-shirts she should sell to maximize her profit.
IIb. Suppose the profit function per week for the restaurant called Benjamin's Burgers is
P(q)= 50 0.5q 500
where q is the number of hamburgers sold.
A. What is the profit if they sell no hamburgers?
B. What is the profit if they sell 900 hamburgers? 901 hamburgers?
C. What is the marginal profit? when x=900?
D. How many hamburgers should they try to sell?
E. What is the profit if they sell 4000 hamburgers? Interpret the sign.
F. Graph the profit function.
IIIb. Josh's Jars business reports the cost and revenue functions shown in the graph below.
A. Mark on the graph the quantity q that would produce a maximum profit.
B. How does the marginal revenue compare to the marginal cost at that point?
C. What is approximately the maximum profit they will make?
D. What is the minimum profit?
IVb. Chris has a business where he sells Chris’s Creative Kites. When selling x kites, the revenue function is: R(x)= 25x and the cost function is: C(x)= .0002x3 + 15x + 1000 dollars.
A)What is the selling price of each kite?
B)What are his fixed costs?
C)How many kites should he sell to maximum his profit?