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At a constant pressure, the temperature \(T\) of an ideal gas is directly proportional to its volume \(V\). (This is Charles’s
Law)
\(T = k V\)
The frequency of a wave \(f\) is inversely proportional to the wavelength of the wave \(\lambda \).
NOTE: The character \(\lambda \) is the lower case Greek letter ‘lambda.’
\(f = \frac {k}{\lambda }\)
The density \(d\) of a material is directly proportional to the mass of the object \(m\) and inversely proportional to its volume
\(V\).
\(d = k \frac {\answer {m}}{\answer {V}}\)
The square of the orbital period of a planet \(P\) is directly proportional to the cube of the semi-major axis of its orbit \(a\). (This is
Kepler’s Third Law of Planetary Motion )
\(P^2 = k a^3\)
The drag of an object traveling through a fluid \(D\) varies jointly with the density of the fluid \(\rho \) and the square of the velocity of the
object \(\nu \).
NOTE: The characters \(\rho \) and \(\nu \) are the lower case Greek letters ‘rho’ and ‘nu,’ respectively.
\(D = k \rho \nu ^2\)
Suppose two electric point charges, one with charge \(q\) and one with charge \(Q\), are positioned \(r\) units apart. The electrostatic
force \(F\) exerted on the charges varies directly with the product of the two charges and inversely with the square of the distance
between the charges. (This is Coulomb’s Law)
\(F = \frac {kqQ}{r^2}\)
NOTE: Note the similarity to this formula and Newton’s Law of Universal Gravitation as discussed in Example gravitylaw.
According to this webpage, the frequency \(f\) of a vibrating string is given by \(f = \frac {1}{2L} \sqrt {\frac {T}{\mu }}\) where \(T\) is the tension, \(\mu \) is the linear mass (Also known as the linear density. It is simply a measure of mass per unit length.) of the string and \(L\) is the length of the
vibrating part of the string. Express this relationship using the language of variation.
Rewriting \(f = \frac {1}{2L} \sqrt {\frac {T}{\mu }}\) as \(f =\frac {1}{2} \frac {\sqrt {T}}{L \sqrt {\mu }}\) we see that the frequency \(f\) varies directly with the square root of the tension and varies inversely with the
length and the square root of the linear mass.
According to the Centers for Disease Control and Prevention www.cdc.gov, a person’s Body Mass Index \(B\) is directly
proportional to his weight \(W\) in pounds and inversely proportional to the square of his height \(h\) in inches.
Express this relationship as a mathematical equation.
\(B = k \frac {\answer {W}}{\answer {h^{2}}}\)
If a person who was \(5\) feet, \(10\) inches tall weighed 235 pounds had a Body Mass Index of 33.7, what is the value
of the constant of proportionality?
\(k = 702.68\)
NOTE: The CDC uses 703.
Rewrite the mathematical equation found in part BMIfirst to include the value of the constant found in part BMIsecond and then find your
Body Mass Index.
\(B = \frac {702.68W}{h^{2}}\)
This exercise refers back to the volume of a right circular cone formula found in Example variationexample.
First assume that \(V\), \(h\) and \(r\) are all measured using the same unit of length. Work with your classmates to show
that in this case, the \(k\) needed for the volume formula \(V = k h r^{2}\) has no units on it.
Now assume that \(V\) is measured in milliliters, \(h\) is measured in meters and \(r\) is measured in yards. Work with your
classmates to find the units on \(k\) so that the volume formula \(V = k h r^{2}\) makes sense.
We know that the circumference of a circle varies directly with its radius with \(2\pi \) as the constant of proportionality. (That is, we
know \(C = 2\pi r.\)) With the help of your classmates, compile a list of other basic geometric relationships which can be seen as variations.
Research the Ideal Gas Law \(PV = nRT\) to see what sorts of units are used for the constant \(R\). What other formulations of this law did
you find in your research?
