MAC2233 (5.5) MDC
5.5 Additional Applications to Business and Economics
The useful life of a machine is the time interval T when revenue R’(t) accumulates at a greater rate than costs
C’(t); thus a manager might dispose of a machine when R’(t) = C’(t).
1. Suppose that a machine is t years old it is generating revenue at a rate R’(t) = 5000 – 20t2 dollars a year and
that operating and servicing costs related to the machine are accumulating at the rate C’(t) = 2,000 + 5t2
dollars a year.
[a] Find the machine’s useful life T. [b] Compute the net profit generated by the machine
over its useful life.
Future value of an income stream: If income f(t) is transferred continuously into an account in which it earns
interest r over a specified period of time 0 ≤ t ≤ T, then the future value of the income stream is the total of all
the deposits plus interest: FV e rT
f (t )e rt dt
2. Money is continuously transferred into a retirement account at a constant rate of $2,000 a year. The
account earns interest at the annual rate of 8% compounded continuously. How much will be in the
account at the end of 25 years?
The present value of an income stream is the amount of money that must be invested now to generate the
same income as the income stream over the same T-year period: PV 0
f (t )e rt dt .
3. A 10-year franchise is being sold. It is expected that t years from now the franchise will be generating
profit at the rate of f(t) = $10,000. If the prevailing annual interest rate of 4% compounded continuously
remains fixed, what is the present value of the franchise?
MAC2233 (5.5) MDC
A function p = D(q) giving the price per unit that consumer’s are willing to spend to get the qth unit of a commodity is
known as the consumer’s demand function. The amount that consumers are willing to pay for q0 units is given by
D( q ) dq is called the total willingness to spend. Consumers’ surplus = total amount consumers would be will to
spend, less what they actually spent: CS 0
D(q) dq p0 q0 , where p0 and q0 denote the actual market price and
corresponding demand, respectively. Producers’ surplus is the difference between what producers would be willing to
accept for supplying q0 units and the price they actually received. If q0 units are sold at a price of p0 and p = S(q) is the
S q dq . Equilibrium is when the demand function and supply
producers’ supply function, then PS p0 q0 0
function are equal.
4. Find the total amount of money consumers are willing to spend if D(q) = and q0 = 10.
5. Find the price p0 = D(q0) at which q0 will be demanded and compute the consumers’ surplus:
D(q) = 150 – 2q – 3q2; q0 = 6
6. Find the price p0 = S(q0) at which q0 will be supplied and compute the producers’ surplus:
S(q) = 0.5q + 15; q0 = 5
7. If D(q) = 245 2q and S(q) = 5 + q, find:
a. the equilibrium price pe (where supply equals demand).
b. consumers’ surplus and producers’ surplus.
MAC2233 (5.5) MDC
5.6 Additional applications to the Life and Social Sciences
Survival and Renewal: Suppose that a population initially has P0 members and that new members
are added at a (renewal) rate of R individuals per year (month, day, hour, etc.). Further suppose that
the fraction of the population that remain for at least t years is given by the (survival) function S(t).
Then T years from now the population will be P T P0 S T RS(T t )dt .
1. P0 = 800,000; R = 500; S(t) = e-0.005t, t in months. Find the population at the end of five months.
2. Polls for a political candidate indicate that the fraction of those who support her t weeks after first
learning of her candidacy is given by f(t) = e-0.03t. At the time she declared her candidacy, 25,000 people
supported her, and new converts are being added at the constant rate of 100 people per week.
Approximately how many people are likely to vote for her if the election is held 20 weeks from the day
she entered the race.
3. A certain nuclear power plant produces radioactive waste at the constant rate of 500 pounds per
year. The waste decays exponentially with a half-life of 28 years. How much of the radioactive
waste will be present after 140 years?
MAC2233 (5.5) MDC
Flow of Blood through an Artery is approximated by 2πrS r dr , where R is the radius of the
artery and S(r) gives the speed of the flow at a distance r from the central axis.
4. Calculate the rate in cubic centimeters per second) at which blood flows through an artery of
radius 0.1 cm if the speed of the blood r cm from the central axis is 8 – 800r2 cm per second.
Cardiac Output: R T0
, where D is the amount of dye injected, C(t) is the concentration of
dye at time t, T0 is the total time required for all the dye to pass the monitoring point, and R is the
cardiac output (liters/min).
5. A physician injects 5 mg of dye into a vein near the heart of a patient and by monitoring the
concentration of the dye in the blood over a 24-second period, determines that the concentration
of dye leaving the heart after t seconds (0 ≤ t ≤ 24) is given by the function
0, 0 t 2
C t . Find the patient’s cardiac output.
0.034(t 26t 48), 2 t 24
6. In a certain undeveloped country, the life expectancy of a person t years old is L(t) years, where
L(t) = 41.6(1 + 1.07t)0.13.
a. Find the life expectancy of a person in this country at age 50.
b. What is the average life expectancy of all people in this country between the ages of 10 and 70?