James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 8.5 Probability 573
of movie goers will increase by 50 per night. Find the demand
function and calculate the consumer surplus when the tickets
are priced at $8.00.
13. If the amount of capital that a company has at time t is f std,
then the derivative, f 9std, is called the net investment flow. Suppose
that the net investment flow is st million dollars per year
(where t is measured in years). Find the increase in capital (the
capital formation) from the fourth year to the eighth year.
14. If revenue flows into a company at a rate of
fstd − 9000s1 1 2t , where t is measured in years and
f std is measured in dollars per year, find the total revenue
obtained in the first four years.
15. If income is continuously collected at a rate of f std dollars per
year and will be invested at a constant interest rate r (compounded
continuously) for a period of T years, then the future
value of the income is given by y T 0 f std ersT2td dt. Compute
the future value after 6 years for income received at a rate of
f std − 8000 e 0.04t dollars per year and invested at 6.2% interest.
16. The present value of an income stream is the amount that
would need to be invested now to match the future value as
described in Exercise 15 and is given by y T 0 f std e2rt dt. Find the
present value of the income stream in Exercise 15.
17. Pareto’s Law of Income states that the number of people with
incomes between x − a and x − b is N − y b a Ax2k dx, where A
and k are constants with A . 0 and k . 1. The average income
of these people is
Calculate x.
x − 1 N yb Ax 12k dx
a
18. A hot, wet summer is causing a mosquito population explosion
in a lake resort area. The number of mosquitoes is increasing at
an estimated rate of 2200 1 10e 0.8t per week (where t is measured
in weeks). By how much does the mosquito population
increase between the fifth and ninth weeks of summer?
19. Use Poiseuille’s Law to calculate the rate of flow in a small
human artery where we can take − 0.027, R − 0.008 cm,
l − 2 cm, and P − 4000 dynesycm 2 .
20. High blood pressure results from constriction of the arteries.
To maintain a normal flow rate (flux), the heart has to pump
harder, thus increasing the blood pressure. Use Poiseuille’s
Law to show that if R 0 and P 0 are normal values of the radius
and pressure in an artery and the constricted values are R and P,
then for the flux to remain constant, P and R are related by the
equation
P
P 0
−S
R0
4
RD
Deduce that if the radius of an artery is reduced to three-fourths
of its former value, then the pressure is more than tripled.
21. The dye dilution method is used to measure cardiac output with
6 mg of dye. The dye concentrations, in mgyL, are modeled by
cstd − 20te 20.6t , 0 < t < 10, where t is measured in seconds.
Find the cardiac output.
22. After a 5.5-mg injection of dye, the readings of dye concentra
tion, in mgyL, at two-second intervals are as shown in the
table. Use Simpson’s Rule to estimate the cardiac output.
t cstd t cstd
0 0.0 10 4.3
2 4.1 12 2.5
4 8.9 14 1.2
6 8.5 16 0.2
8 6.7
23. The graph of the concentration function cstd is shown after
a 7-mg injection of dye into a heart. Use Simpson’s Rule to
estimate the cardiac output.
y
(mg/L)
6
4
2
0
2 4 6 8 10 12 14
t (seconds)
Calculus plays a role in the analysis of random behavior. Suppose we consider the
choles terol level of a person chosen at random from a certain age group, or the height
of an adult female chosen at random, or the lifetime of a randomly chosen battery of a
certain type. Such quantities are called continuous random variables because their
values actually range over an interval of real numbers, although they might be measured
or recorded only to the nearest integer. We might want to know the probability that a
blood cholesterol level is greater than 250, or the probability that the height of an adult
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