ECON 143-144 CALCULUS SELECTED EXERCISE PROBLEMS
1) Break-even analysis. A publisher for a promising new novel figures fixed costs
(overhead, advances, promotion, copy editing, typesetting) at $55,000, and variable costs
(printing, paper, binding, shipping) at $1.60 for each book produced. If the book is sold to
distributors for $11 each, how many must be produced and sold for the publisher to break
even?
2) Equipment rental. The local supermarket rents carpet cleaners for $20 a day. These
cleaners use shampoo in a special cartridge that sells for $16 and is available only from the
supermarket. A home carpet cleaner can be purchased for $300. Shampoo for the home
cleaner is readily available for $9 a bottle. Past experience has shown that it takes two
shampoo cartridges to clean the 10-foot-by-12- foot carpet in your living room with the rented
cleaner. Cleaning the same area with the home cleaner will consume three bottles of shampoo.
If you buy the home cleaner, how many times must you clean the living-room carpet to make
buying cheaper than renting?
3) Business Markup policy. A clothing store sells a shirt costing $20 for $33 and a jacket
costing $60 for $93.
(A) If the markup policy of the store is assumed to be linear, write an equation that expresses
retail price R in terms of cost C (wholesale price).
(B) What does a store pay for a suit that retails for $240?
4) Business Depreciation. A charter fishing company buys a new boat for $224,000 and
assumes that it will have a trade in value of $115,200 after 16 years.
(A) Find a linear model for the depreciated value V of the boat t years after it was purchased.
(B) What is the depreciated value of the tractor after 10 years?
(C) When will the depreciated value fall below $100,000?
(D) Graph V for and illustrate the answers from (A) and (B) on the graph.
5) Supply and demand. At a price of $2.28 per bushel, the supply of barley is 7,500 million
bushels and the demand is 7,900 million bushels. At a price of $2.37 per bushel, the supply is
7,900 million bushels and the demand is 7,800 million bushels.
(A) Find a price supply equation of the form p= mx + b
(B) Find a price demand equation of the form p= mx + b
(C) Find the equilibrium point.
(D) Graph the price supply equation, price demand equation, and equilibrium point in the
same coordinate system
6) Cost analysis. A plant can manufacture 50 tennis rackets per day for a total daily cost of
$3,855 and 60 tennis rackets per day for a total daily cost of $4,245.
(A) Assuming that daily cost and production are linearly related, find the total daily cost of
producing x tennis rackets.
(B) Graph the total daily cost for 0 ≤ x ≤ 100
(C) Interpret the slope and y intercept of this cost equation.
7) Maximum Revenue the financial department in the company that produces a digital
camera arrived at the following price demand function and the corresponding revenue
function:
p(x) = 94,8 -5x Price demand function
R(x) = xp(x) = x(94.8 - 5x) Revenue function
where p(x) is the wholesale price per camera at which x million cameras can be sold and R(x)
is the corresponding revenue (in millions of dollars). Both functions have domain 1 ≤ x ≤ 15.
(A) Find the value of x to the nearest thousand cameras that will generate the maximum
revenue. What is the maximum revenue to the nearest thousand dollars? Solve the problem
algebraically by completing the square.
(B) What is the wholesale price per camera (to the nearest dollar) that generates the maximum
revenue?
8) Revenue. The marketing research department for a company that manufactures and sells
notebook computers established the following price demand and revenue functions:
where p(x) is the wholesale price in dollars at which x thousand computers can be sold, and
R(x) is in thousands of dollars. Both functions have domain 1 ≤ x ≤ 25
(A) Sketch a graph of the revenue function in a rectangular coordinate system.
(B) Find the value of x that will produce the maximum revenue. What is the maximum
revenue to the nearest thousand dollars?
(C) What is the wholesale price per computer (to the nearest dollar) that produces the
maximum revenue?
9) Solve the equations
4x + y + 3z = 8
-2x + 5y + z = 4
3x + 2y + 4z = 9
10) Solve the following system of equations:
2x + 2y - 5z = -5
x -y + z = 3
-3x + y + 2z = -2
Solve the quadratic equations
(a) 2x2 + 9x + 5 = 0
(b) x2 - 4x + 4 = 0
(c) 3x2 - 5x + 6 = 0
11) Give a rough sketch of the graph of the following quadratic function:
f(x) = -x2 + 8x - 12
12) Use the three-step strategy to produce rough graphs of the following quadratic functions:
(a) f (x) = 2x2 - 11x - 6 (b) f (x) = x2 - 6x + 9
13) Given the supply and demand functions
P =Q2S + 14Q2S + 22
P = -Q2D - 10QD + 150
calculate the equilibrium price and quantity. And sketch the graph
14) Given the supply and demand functions
P = 2Q2S + 10QS + 10
P = -Q2D -5QD + 52
calculate the equilibrium price and quantity. And sketch the graph
15)Given the quadratic supply and demand functions
P = Q2s+ 2QS + 12
P = -Q2D - 4QD + 68 determine the equilibrium price and quantity. And sketch the graph.
16) Given the supply and demand functions
P = Q2S + 2QS + 7
P = -QD + 25
determine the equilibrium price and quantity.
17) Given the demand function
P = 100 - 2Q
express TR as a function of Q and hence sketch its graph.
(a) For what values of Q is TR zero?
(b) What is the maximum value of TR?
18) If fixed costs are 4, variable costs per unit are 1 and the demand function is
P = 10 - 2Q
obtain an expression for π in terms of Q and hence sketch a graph of π against Q.
(a) For what values of Q does the firm break even?
(b) What is the maximum profit?
19) Since the beginning of the year, the price of a bottle of soda at a local discount
supermarket has been rising at a constant rate of 2 cents per month. By November first, the
price had reached $1.56 per bottle. Express the price of the soda as a function of time and
determine the price at the beginning of the year.
20) CREDIT CARD DEBT A credit card company estimates that the average cardholder
owed $7,853 in the year 2000 and $9,127 in 2005. Suppose average cardholder debt D grows
at a constant rate.
a. Express D as a linear function of time t, where t is the number of years after 2000. Draw the
graph.
b. Use the function in part (a) to predict the average cardholder debt in the year 2010.
c. Approximately when will the average cardholder debt be double the amount in the year
2000?
21) MANUFACTURING COST A manufacturer estimates that it costs $75 to produce each
unit of a particular commodity. The fixed overhead is $4,500. Express the total cost of
production as a function of the number of units produced and draw the graph.
22) LINEAR DEPRECIATION A doctor owns $1,500 worth of medical books which, for tax
purposes, are assumed to depreciate linearly to zero over a 10-year period. That is, the value
of the books decreases at a constant rate so that it is equal to zero at the end of 10 years.
Express the value of the books as a function of time and draw the graph.
A manufacturer can sell a certain product for $110 per unit. Total cost consists of a fixed
overhead of $7,500 plus production costs of $60 per unit.
a. How many units must the manufacturer sell to break even?
b. What is the manufacturer’s profit or loss if 100 units are sold?
c. How many units must be sold for the manufacturer to realize a profit of $1,250?