7.8 Applications Using Rational Equations and Proportions
Formulas
Direct Variation
Inverse Variation
Joint Variation and Combined Variation
Models
Formulas
Ex. 1 The following is a formula used by electricians to determine the
resistance R of two resistors r1 and r2 connected in parallel.
Solve for r1 .
!
"
=
!
$%
+
!
$'
To solve a rational equation for a Specified Variable:
1)
2)
3)
4)
5)
Multiply both sides by the LCM (to clear fractions),
Remove parentheses, if necessary,
Get all terms of the specified variable alone on one side,
Factor out the specified variable,
Multiply or divide both sides to isolate the variable.
Ex. 2 The formula below is used to find a satellite’s escape velocity V,
where R is a planet’s radius, h is the satellite’s height above the planet,
and g is the planet’s gravitational constant. Solve for h.
𝑉)
2𝑔
=
𝑅) 𝑅 + ℎ
2
Ex. 3 The following formula is used to determine the frequency f of a
sound that is moving at velocity v toward a listener who hears the
sound as frequency g. Here, s is the speed of sound in a particular
medium. Solve for s.
𝑓=
𝑠𝑔
𝑠+𝑣
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Variation
A fitness trainer earns $22 per hour.
This is a direct variation (varies directly) because the earnings divided
by the hours worked will always be $22. In this case the 22 is a
constant of variation.
1
2
= 22𝑜𝑟𝐸 = 22𝑡The earnings vary directly as the time worked.
When f(x) = kx
or y = kx
there is a direct variation.
We say “y varies directly as x”, or y is proportional to x. where k is the
constant of variation or constant of proportionality.
Ex. 4
Find the variation constant and an equation of variation if
y varies directly as x, and y = 32 when x = 2.
(𝑦 𝑥 = 16𝑥𝑜𝑟𝑓 𝑥 = 16𝑥 is also used.)
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Ex. 5 The speed v of a train of ocean waves varies directly as the swell
period t, or time between consecutive waves. Waves with a swell period
of 12 sec. are traveling 21 mph. How fast are waves traveling that have
a swell period of 20 sec.?
Inverse Variation
inverse variation.
When
<
<
=
=
𝑓 𝑥 = 𝑜𝑟𝑦 =
this is an
We say “y varies inversely as x”, or “y is inversely proportional to x.”
k is the constant of variation or constant of proportionality
Ex. 6 Find the variation constant and an equation of variation if
y varies inversely as x, and y = 32 when x = 0.2.
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Ex. 7 The time t that it takes to download a movie file varies inversely
as the transfer speed s of the Internet connection. A typical fulllength movie file will transfer in 48 min. at a transfer speed of 256KB/s
(kilobytes per second).
How long will it take to transfer the same movie file at a transfer speed
of 32 KB/s?
Joint Variation
When there is a direct variation with more than one
variable we say there is a “joint variation” or “varies jointly”.
We write
𝑦 = 𝑘𝑥𝑧 y varies jointly as x and z.
Example 8
Find an equation of variation if y varies jointly as x and z,
and
y = 30, when x = 2 and z = 3.
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Combined Variations may have direct and inverse variations with more
than one variable.
Ex. 9 Find a equation of variation if y varies jointly as x and z and
inversely as the square of w, and y = 105 when x = 3, z = 20, and w = 2.
Note:
For k > 0
𝒚 = 𝒌𝒙
For k > 0
𝒚=
𝒌
𝒙
as x increases,
as x increases
y increases as well.
y decreases
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Ex. 10 In general, animals with higher heart rates have shorter life
pans. The table blow lists average heart rates in number of beats per
minute, and average life spans, in years.
a) Determine whether the data indicate direct or inverse variation.
b) Use the data point (70, 25) to find an equation that describes the
data.
c) Use the equation to estimate the life span of a cat with a heart
rate of
150 beats per minute.
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