Looking for a Unsorted Economics Worksheet Templates? Download it for free!

Math 105- Calculus for Economics & Business

Sections 10.3 & 10.4 : Optimization problems

How to solve an optimization problem?

1. Step 1: Understand the problem and underline what is important ( what is known, what is unknown,

what we are looking for, dots)

2. Step 2: Draw a “diagram”; if it is possible.

3. Step 3: Assign “symbols” or “variables” for all the quantities involved (know or unknown), and label

the diagram.

4. Step 4: Write the quantity Q to be maximized or minimized in terms of some of the previous variables

(from Step 3). Example: Q = g(x, y, h)

5. Step 5: Rewrite Q as a function of only one variable. To do this, ﬁnd the relationships between the

variables using the information given in the problem. Then, use these equations to eliminate all but one

of the variables in the expression of Q. Thus, we get Q = f (x).

6. Step 5: Use the methods of sections 10.1 and 10.2 to ﬁnd the maximum or the minimum of the quantity

Q = f (x).

7. REMARK: Do not forget to ﬁnd the endpoints and to check if the maximum or the minimum is at these

points if you have more than one critical number in the domain.

8. Short-cut: If there is only one critical number a in the domain, then :

if we have a local maximum at a =

we have a global maximum at a,

if we have a local minimum at a

we have a global minimum at a.

In other term, we do not need to check the position of the endpoints.

9. Reminder: At the worksheet I gave you in the beginning of the semester (it is the KEY FORMULAS for

Chapter 9 posted at the homework assignment web page) of the textbook, you can ﬁnd all the formulas

related to the cost, revenue and proﬁt. You should work

Examples:

Problem 1. The regular air fare between Boston and San Francisco is $500. An airline using

planes with a capacity of 300passengers on this route observes that they ﬂy with an average of

180 passengers. Market research tells the airlines’ managers that each $ 5 fare reduction would

attract, on average, 3 more passengers for each ﬂight. How should they set the fare to maximize

their revenue? Explain your reasoning to receive credit.

Let R= the revenue function = quantity

price

Let n= the number of times the fare is reduced by $ 5 dollars. Then:

price

= $500

n ($5) = 500

5n dollars,

quantity = number of passengers = 180passengers + n (3passengers) = 180 + 3n passengers.

Hence

R(n) = (180 + 3n)(500

5n) = 90 000 + 600n

15n

to maximize (for 0

40).

600

R (n) = 600

30n = 0

n =

= 20 is the only critical number.

R (n) =

R (20) =

30 < 0. By the second derivative test, R has a local maximum at n = 20,

which is an absolute maximum since it is the only critical number.

The best fare to maximize the revenue is then: $ 500

5(20) = $400 , with 180 + 3(20) = 240 passengers

and a revenue of R(20) = $96, 000 .

Optimization Problems Worksheet - Math 105, Sections 10.3 & 10.4 Calculus For Economics & Business

Related Articles

Related forms

Related Categories