Chapter Twenty-One — Rational Equations and Inequalities
The Humongous Book of Algebra Problems
475
Direct and Indirect Variation
Turn a word problem into a rational equation
21.21 Assume that the value of x varies directly with the value of y according to
the constant of proportionality k. Identify two equations that describe the
relationship between x and y.
If x and y vary proportionally, then y = kx and .
Note: Problems 21.22–21.23 refer to the direct variation relationship described below.
21.22 A professional sports team notes that the ambient crowd noise at home
games is directly proportional to the attendance at those games. During one
game, the cheering of a maximum capacity crowd of 75,000 fans averages 90
decibels. Identify the constant of proportionality k.
Let a represent attendance and n represent the noise (in decibels) generated by
that population. If n varies directly with a, then (according to Problem 21.21)
n = ka. Substitute n = 90 and a = 75,000 into the equation and solve for k.
Note: Problems 21.22–21.23 refer to the direct variation relationship described in Problem
21.22.
21.23 Approximately how loud is a crowd of 62,000 fans?
Substitute a = 62,000 and k = 0.0012 into the variation equation n = ka to
calculate n.
Varying
proportionally means
the same thing as
varying directly.
Let’s say
k = 2. According
to the equation
y = kx, y is always twice
as big as x. Because
y is always two times
as large as x, y ÷ x
always equals 2.
This value
of k (and the
equation n = ka
that you plug it into)
come from Problem
21.22.
Chapter Twenty-One — Rational Equations and Inequalities
The Humongous Book of Algebra Problems
476
21.24 Assume x and y vary proportionally. If x = 16 when y = –3, calculate the value of
y when x = 6.
If x and y vary proportionally, then x = ky. Substitute x = 16 and y = –3 into the
equation and solve for k.
To determine the value of y when x = 6, substitute x = 6 and into the
proportionality equation.
21.25 Assume that the growth of a vine is directly proportional to the time it is
exposed to light. If the vine is exposed to 72 hours of light and grows 1.5
inches, how long will the vine grow when exposed to 200 hours of light?
Round the answer to the hundredths place.
Let l represent the length the vine grows when exposed to h hours of light. The
values vary proportionally, so l = kh, where k is a constant of proportionality.
Substitute l = 1.5 and h = 72 into the equation to calculate k.
To determine how long the vine grows after 200 hours of light, substitute
h = 200 and into the equation l = kh.
The vine grows approximately 4.17 inches when exposed to 200 hours of light.
To cancel out
the coefcient of
y, multiply both sides
of the equation by
its reciprocal.
Rounding
4.166666… to
the hundredths
place gives you
4.17.
Chapter Twenty-One — Rational Equations and Inequalities
The Humongous Book of Algebra Problems
477
21.26 Assume that the value of x varies inversely with the value of y according to the
constant of variation k. Identify two equations that describe the relationship
between x and y.
If x and y vary inversely, then and xy = k.
Note: Problems 21.27–21.28 refer to the inverse variation relationship described below.
21.27 A computer security company determines that the chance a computer is
infected with a virus is inversely proportional to the number of times the virus
protection software is updated during the year.
If a user updates the software 250 times in one year, there is a 1.25% chance
that the system will become infected with a virus. Calculate the corresponding
constant of variation k.
Let u represent the number of updates applied in one year and v represent
the corresponding chance that the computer will be infected during that
year. Because v varies inversely with u, uv = k (according to Problem 21.26).
Substitute u = 250 and v = 0.0125 into the equation to calculate k.
Note: Problems 21.27–21.28 refer to the inverse variation relationship described in
Problem 21.27.
21.28 What are the chances a computer will be infected with a virus during a year in
which the antivirus software is updated once a month? Round the answer to
the nearest whole percentage.
Substitute u = 12 and k = 3.125 into the equation uv = k and solve for v.
There is a 26% chance the computer will be infected with a virus during that
year.
Inverse
variation is also
called indirect
variation.
If x and
y vary in-
versely, then
the product of x
and y is constant.
For example, let’s
say x = 2 and y = 5,
so xy = 10. If y gets
bigger (y = 10) then
x has to get smaller
(x = 1) to keep the
product constant
(1
˙
10 = 10).
You can
also refer to
the constant of
proportionality in
direct variation
problems as the
constant of
variation.
The value of v
should be a percen-
tage converted into
a decimal, so drop the
percentage sign and
move the decimal two
places to the left:
v = 1.25% = 0.0125.
Convert the decimal into
a percentage by moving the decimal
point two places to the right:
0.260416 = 26.0416%.
Chapter Twenty-One — Rational Equations and Inequalities
The Humongous Book of Algebra Problems
478
21.29 Assume x and y vary inversely. If x = –9 when y = 4, calculate the value of x
when y = –18.
If x and y vary inversely, then xy = k, where k is the constant of variation.
Substitute x = –9 and y = 4 into the equation to calculate k.
To determine the value of x when y = –18, substitute y = –18 and k = –36 into the
equation xy = k and solve for x.
21.30 A police commissioner determines that the number of robberies committed in
a downtown tourist district is inversely proportional to the number of police
officers assigned to patrol the area on foot.
In June, 80 officers were assigned patrols and 35 robberies were reported. If
budget cuts to the city’s budget dictate that only 65 officers will be assigned in
July, approximately how many robberies will occur during that month? Round
the answer to the nearest whole number.
Let p represent the number of officers on patrol and r represent the corre-
sponding number of robberies during a given month. Because p and r are
inversely proportional, pr = k, where k is the constant of variation. Substitute
p = 80 and r = 35 into the equation to calculate k.
To predict the number of robberies in July, substitute k = 2,800 and p = 65 into
the equation pr = k and solve for r.
Approximately 43 robberies will occur during July.
The words
inversely proportional
indicate inverse
variation.
..................Content has been hidden....................
You can't read the all page of ebook, please click here login for view all page.