Mathematical formulas that involve exponents logarithms powers and roots have constraints on the values that can be assigned or associated with two or more variables Realworld formulas involving measurements ID: 381066
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IntroductionMathematical formulas that involve exponents, logarithms, powers, and roots have constraints on the values that can be assigned or associated with two or more variables. Real-world formulas involving measurements like area, length, time, and volume have additional restraints, whereas others (such as current, direction, and voltage) can have positive and negative values in addition to the limits imposed by real-world relationships.
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6.3.3 Rearranging FormulasSlide2
Introduction, continuedRearranging variables in a formula can result in changes to these constraints, so the result of rearranging variables has to be checked and redefined in those cases. Formulas can often be analyzed and visualized in the same way that equations and functions can, and the data resulting from formulas can be presented as data in tables or in graphs.
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6.3.3 Rearranging FormulasSlide3
Key ConceptsA formula relates two or more variables in a mathematical or a real-world problem context. The values of the variables in a formula can be constrained or limited by mathematical or real-world conditions.
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6.3.3 Rearranging FormulasSlide4
Key Concepts, continuedFor example, the formula for the area of a triangle contains three quantities that are positive. If a triangle is graphed on a coordinate plane and its orientation on the plane results in either b or h being assigned a negative value, then the other variable must also be assigned a negative value so that the area of the triangle is positive.
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6.3.3 Rearranging FormulasSlide5
Key Concepts, continuedRearranging a formula can leave the constraints on variable values the same or it can change them. For example, in the distance-rate-time formula d = rt, time is constrained by the condition t ≥ 0, with t = 0 signifying the time at which motion begins. The
formula can be rewritten as ;
however,
is
undefined at
t
= 0, so in this case
t
> 0. This corresponds to the real-world condition that r ≠ 0 only when t ≠ 0, since at t = 0, there is no distance covered
and no motion.
56.3.3 Rearranging FormulasSlide6
Key Concepts, continuedFormulas can be rearranged to express one or more variables in terms of another variable. A rearranged formula can reduce the number of variables to be calculated.6
6.3.3 Rearranging FormulasSlide7
Common Errors/Misconceptionsusing the wrong operation (e.g., dividing instead of multiplying) in the process of isolating a variable in a formulaforgetting to check that the constraints on a variable are the same after the variable is isolated or relocated in a formula
confusing a mathematical constraint with a real-world constraint in a formula
forgetting
to check both the mathematical
and
real-world constraints on the variable(s)
in a
rearranged formula
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6.3.3 Rearranging FormulasSlide8
Guided PracticeExample 1The pressure and temperature inside an insulated hot-beverage bottle is related to the volume of the bottle and the amount of beverage in it by a real-world form of the ideal gas law. The ideal gas law is given by the formula PV = nRT, in which n is the number of moles (a unit of
counting) of the gas in a container, P is the pressure the gas exerts on the container, V
is the volume of the container, and
T
is
the temperature in degrees Kelvin. The only constant in the formula is
R
. Rearrange the formula to show how the temperature
T is affected by doubling each variable n, P, and V
. (Note: All of the quantities in the formula are nonzero.)
86.3.3 Rearranging FormulasSlide9
Guided Practice: Example 1, continuedIsolate temperature, T, in the given formula, PV =
nRT.
Use division to isolate
T
on one side of the equation.
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6.3.3 Rearranging FormulasSlide10
Guided Practice: Example 1, continued
The formula PV = nRT, isolated for
T
,
is
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6.3.3 Rearranging Formulas
Simplify.
Apply the Symmetric Property of Equality.
Given formula
Divide both sides of the formula by
nR
.Slide11
Guided Practice: Example 1, continuedDetermine how T is affected in the rearranged formula if n is doubled and
P and V
stay the same
.
We are given that
R
is a constant, so it will not be affected by
changes to
the other variables.Let the original value of n be n1
. If n1 is doubled, the resulting value of
n is 2n1.Write the revised ideal gas law from step 1 for both conditions.
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6.3.3 Rearranging FormulasSlide12
Guided Practice: Example 1, continued Formula rewritten for n1 (the original value of n):Formula
rewritten for 2n1 (twice the original value of n)
:
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6.3.3 Rearranging FormulasSlide13
Guided Practice: Example 1, continued Notice that T2 is half of T
1. In other words, doubling the value of n reduces
the temperature by
half
. In general, increasing
n
will
decrease
T
if P and V stay the same.
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6.3.3 Rearranging FormulasSlide14
Guided Practice: Example 1, continuedDetermine how T is affected in the rearranged formula if P is doubled and
n and V stay the same
.
Let the original value of
P
be
P
1
. If P1 is doubled, the resulting value of P is 2P1.
Write the revised ideal gas law from step 1 for both conditions.Formula rewritten for P1
(the original value of P):14
6.3.3 Rearranging FormulasSlide15
Guided Practice: Example 1, continued Formula rewritten for 2P1 (twice the original value of P):
Notice that T2 is twice
T
1
. In other words,
doubling
the value of
P
doubles the temperature. In general, increasing P increases T if n and
V stay the same.
156.3.3 Rearranging FormulasSlide16
Guided Practice: Example 1, continuedDetermine how T is affected in the rearranged formula if V is doubled and
n and P
stay the same
.
Note
that
V
, like
P, is a factor of the numerator of the formula
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6.3.3 Rearranging FormulasSlide17
Guided Practice: Example 1, continuedSince both V and P are in the numerator, changes in the volume, V, would have the same effect on temperature, T, as changes in pressure, P. In other words, if we were to double the volume instead of the pressure, the temperature would still be doubled.
Therefore, increasing V
increases
T
if
n
and
P
stay the same.
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6.3.3 Rearranging Formulas
✔Slide18
Guided Practice: Example 1, continued186.3.3 Rearranging FormulasSlide19
Guided PracticeExample 3The formula for a standard earthquake-body wave scale, mb, is given by , in which
A is the amplitude
of the ground motion in microns (10
–6
meter),
T
is the period of the wave, and
Q
is
a correction constant. Determine a formula for the frequency of the earthquake wave if the frequency F is defined as the reciprocal of the wave period
. Rearrange the formula to find the range of F values for when the range of T
values is [4, 5] seconds. Then, rearrange the formula to find the range of values of A for when the range of values of mb is [6, 9], the range of T values is [4, 5] seconds, and Q = 2.
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6.3.3 Rearranging FormulasSlide20
Guided Practice: Example 3, continuedSubstitute an expression for F in place of T in the formula
.
Frequency
F
can be written as
.
The argument
of the logarithm can
be written as
. Therefore, the argument of the logarithm can be written as
AF.
206.3.3 Rearranging FormulasSlide21
Guided Practice: Example 3, continued
The formula, using
F
in place of
T
, is
m
b
= log (AF) + Q.
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6.3.3 Rearranging Formulas
Original formula
Substitute
AF forSlide22
Guided Practice: Example 3, continuedUse the definition of the logarithm to rewrite the formula with an exponential term. Then, solve it for F.
Isolate the logarithm from the rest of the formula, then simplify the result and solve it for F.
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6.3.3 Rearranging FormulasSlide23
Guided Practice: Example 3, continued
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6.3.3 Rearranging Formulas
Definition of the logarithm
Simplify the exponential term.
Divide both sides by
A
.
m
b
= log (
AF
) +
Q
Modified formula
m
b
–
Q
= log (
AF
)
Subtract
Q
from both sides.Slide24
Guided Practice: Example 3, continuedUse the relationship between the period and the frequency to calculate the range of frequency values. The period
T is related to the frequency F by the
relationship
If the range of
T
is [4, 5] seconds, than the range of
F
values is the reciprocal
of the period interval values,
or
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6.3.3 Rearranging FormulasSlide25
Guided Practice: Example 3, continuedWrite the formula for the earthquake-magnitude scale so that the factors with known values are isolated from A.
To find the range of values of A for when the range of values of m
b
is
[6, 9] and
Q
= 2, first substitute 2 for Q
in
the rewritten formula
, , and then rearrange the formula so that it’s in terms of A.
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6.3.3 Rearranging FormulasSlide26
Guided Practice: Example 3, continued266.3.3 Rearranging FormulasSlide27
Guided Practice: Example 3, continuedThe formula, written in terms of A when Q = 2, is
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6.3.3 Rearranging FormulasSlide28
Guided Practice: Example 3, continuedUse the result of the previous step to write the maximum and minimum values of the range of values for A.
Use the formula, to determine the minimum and maximum values using the given ranges of
T
and
m
b
.
The range values for
T
are given as [4, 5], so the minimum value for T is 4 and the maximum value is 5.The range values for mb are given as [6, 9]. Therefore, the
minimum value for mb is 6 and the maximum value is 9
.286.3.3 Rearranging FormulasSlide29
Guided Practice: Example 3, continued Substitute the minimum values, T = 4 and mb = 6:
The minimum value of
A
is 4 • 10
4
, or about
40,000
centimeters
.
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6.3.3 Rearranging Formulas
Formula for the minimum value of
A
Substitute 4 for T and 6 for m
b.
Simplify.Slide30
Guided Practice: Example 3, continued Substitute the maximum values, T = 5 and mb = 9:
The maximum value of A is 5 • 10
7
, or
about
50 meters.
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6.3.3 Rearranging Formulas
✔
Formula for the maximum value of
A
Substitute 5 for
T
and 9 for
m
b
.
Simplify.Slide31
Guided Practice: Example 3, continued316.3.3 Rearranging Formulas