Two variables are directly proportional (or just proportional) if the ratios of their corresponding values are always equal. Consider the functions described in the tables below. The first table shows the price of gasoline as a function of the number of gallons purchased.
The ratio \(\dfrac{\text{total price}}{\text{number of gallons}}\text{,}\) or price per gallon, is the same for each pair of values in the first table. This agrees with everyday experience: The price per gallon of gasoline is the same no matter how many gallons you buy. Thus, the total price of a gasoline purchase is directly proportional to the number of gallons purchased.
The second table shows the population of a small town as a function of the townβs age. The ratio \(\dfrac{\text{number of people}}{\text{number of years}}\) gives the average rate of growth of the population in people per year. You can see that this ratio is not constant; in fact, it increases as time goes on. Thus, the population of the town is not proportional to its age.
We see that the price, \(P\text{,}\) of a fill-up is a linear function of the number of gallons, \(g\text{,}\) purchased. This should not be surprising if we write an equation relating the variables \(g\) and \(P\text{.}\) Because the ratio of their values is constant, we can write
\begin{equation*}
\frac{P}{g}= k
\end{equation*}
where \(k\) is a constant. In this example, the constant \(k\) is \(2.40\text{,}\) the price of gasoline per gallon. Solving for \(P\) in terms of \(g\text{,}\) we have
\begin{equation*}
P = kg = 2.40g
\end{equation*}
which we recognize as the equation of a line through the origin.
The amount of interest, \(I\text{,}\) earned in one year on an account paying 7% simple interest, varies directly with the principal, \(P\text{,}\) invested, because
In the opening example, we saw that the price of gasoline, \(P\text{,}\) is a linear function of the number of gallons purchased. From the definition above, we see that any direct variation defines a linear function of the form
\begin{equation*}
y = f(x) = kx
\end{equation*}
Compareing this equation to the standard form for a linear function, \(y = b + mx\text{,}\) we see that the constant term, \(b\text{,}\) is zero, so the graph of a direct variation passes through the origin. The positive constant \(k\) in the equation \(y = kx\) is just the slope of the graph, so it tells us how rapidly the graph increases.
Subsection5.4.2The Scaling Property of Direct Variation
Because the graph of \(y=kx\) passes through the origin, direct variation has the following scaling property: if we double the value of \(x\text{,}\) then the value of \(y\) will double also. In fact, increasing \(x\) by any factor causes \(y\) to increase by the same factor. Look again at the table for the price of buying gasoline. Doubling the number of gallons of gas purchased, say, from \(4\) gallons to \(8\) gallons or from \(6\) gallons to \(12\) gallons, causes the total price to double also.
Thus, \(T(u)\) is a linear function of \(u\text{,}\) but the \(T\)-intercept is \(400\text{,}\) not \(0\text{.}\) Your tuition is not proportional to the number of units you take, so this is not an example of direct variation. You can check that doubling the number of units does not double the tuition. For example,
Subsection5.4.3Finding a Formula for Direct Variation
If we know any one pair of values for the variables in a direct variation, we can find the constant of variation. Then we can use the constant to write a formula for one of the variables as a function of the other.
If you kick a rock off the rim of the Grand Canyon, its speed, \(v\text{,}\) varies directly with the time, \(t\text{,}\) it has been falling. The rock is falling at a speed of 39.2 meters per second when it passes a lizard on a ledge 4 seconds later.
The surface area of a sphere varies directly with the square of its radius. A balloon of radius 5 centimeters has surface area \(100\pi\) square centimeters, or about 314 square centimeters.
Find a formula \(S=f(r)\) for the surface area of a sphere as a function of its radius \(~\alert{\text{[TK]}}\text{.}\)
The graph has the shape of the basic function \(y=x^2\text{,}\) so all we need are a few points to "anchor" the graph. We know that \(f(5)=314\text{,}\) and \(f(1)= 4\pi \cdot 1^2\) or about \(12.6\text{.}\) A graph is shown below.
The volume of a sphere varies directly with the cube of its radius. A balloon of radius 5 centimeters has volume \(\dfrac{500\pi}{3}\) cubic centimeters, or about 524 cubic centimeters.
Find a formula for the volume, \(V\text{,}\) of a sphere as a function of its radius, \(r\text{.}\)
In any example of direct variation, as the input variable increases through positive values, the output variable increases also. Thus, a direct variation is an increasing function, as we can see when we consider the graphs of some typical direct variations shown below.
The graph of a direct variation always passes through the origin, so when the input is zero, the output is also zero. Thus, the functions \(~y = 3x + 2~\) and \(~y = 0.4x^2 - 2.3~\text{,}\) for example, are not direct variation, even though they are increasing functions for positive \(x\text{.}\)
Even without an equation, we can check whether a table of data describes direct variation or merely an increasing function. If \(y\) varies directly with \(x^n\text{,}\) then \(~y = kx^n~\text{,}\) or, equivalently,\(~\dfrac{y}{x^n} = k.~\)
If \(y\) varies directly with \(x^2\text{,}\) then \(y = kx^2\text{,}\) or \(\dfrac{y}{x^2}= k\text{.}\) Delbert should calculate the ratio \(\dfrac{y}{x^2}\) for each data point.
You probably know that the answer to both of these questions is No. The area of a circle is proportional to the square of its radius, and the volume (and hence the weight) of an object is proportional to the cube of its linear dimension. Variation with a power of \(x\) produces a different scaling effect.
The Taipei 101 building is 1671 feet tall, and in 2006 it was the tallest skyscraper in the world. Show that doubling the dimensions of a model of the Taipei 101 building produces a model that weighs 8 times as much.
The Taipei 101 skyscraper is approximately box shaped, so its volume is given by the product of its linear dimensions, \(V = lwh\text{.}\) The weight of an object is proportional to its volume, so the weight, \(W\text{,}\) of the model is
\begin{equation*}
W = klwh
\end{equation*}
where the constant \(k\) depends on the material of the model. If we double the length, width, and height of the model, then
In general, if \(y\) varies directly with a power of \(x\text{,}\) that is, if \(y = kx^n\text{,}\) then doubling the value of \(x\) causes \(y\) to increase by a factor of \(2^n\text{.}\) In fact, if we multiply \(x\) by any positive number \(c\text{,}\) then
We will call \(n\) the scaling exponent, and you will often see variation described in terms of scaling. For example, we might say that "the area of a circle scales as the square of its radius." (In many applications, the power \(n\) is called the scale factor, even though it is not a factor but an exponent.)
Delbertβs credit card statement lists three purchases he made while on a business trip in the Midwest. His companyβs accountant would like to know the sales tax rate on the purchases.
At constant acceleration from rest, the distance traveled by a race car is proportional to the square of the time elapsed. The highest recorded road-tested acceleration is 0 to 60 miles per hour in 3.07 seconds, which produces the following data.
The weight of an object on the Moon varies directly with its weight on Earth. A person who weighs 150 pounds on Earth would weigh only 24.75 pounds on the Moon.
Find a function that gives the weight \(m\) of an object on the Moon in terms of its weight \(w\) on Earth. Complete the table and graph your function in a suitable window.
Hubbleβs law says that distant galaxies are receding from us at a rate that varies directly with their distance. (The speeds of the galaxies are measured using a phenomenon called redshifting.) A galaxy in the constellation Ursa Major is 980 million light-years away and is receding at a speed of 15,000 kilometers per second.
Find a function that gives the speed, \(v\text{,}\) of a galaxy in terms of its distance, \(d\text{,}\) from Earth. Complete the table and graph your function in a suitable window. (Distances are given in millions of light-years.)
The length, \(L\text{,}\) of a pendulum varies directly with the square of its period, \(T\text{,}\) the time required for the pendulum to make one complete swing back and forth. The pendulum on a grandfather clock is 3.25 feet long and has a period of 2 seconds.
Express \(L\) as a function of \(T\text{.}\) Complete the table and graph your function in a suitable window.
The amount of power, \(P\text{,}\) generated by a windmill varies directly with the cube of the wind speed, \(w\text{.}\) A windmill on Oahu, Hawaii, produces 7300 kilowatts of power when the wind speed is 32 miles per hour.
Express the power as a function of wind speed. Complete the table and graph your function in a suitable window.
A wide pipe can handle a greater water flow than a narrow pipe. The graph shows the water flow through a pipe, \(w\text{,}\) as a function of its radius, \(r\text{.}\) How great is the water flow through a pipe of radius of 10 inches?
The faster a car moves, the more difficult it is to stop. The graph shows the distance, \(d\text{,}\) required to stop a car as a function of its velocity, \(v\text{,}\) before the brakes were applied. What distance is needed to stop a car moving at 100 kilometers per hour?
The maximum height attained by a cannonball depends on the speed at which it was shot. The table shows maximum height as a function of initial speed. What height is attained by a cannonball whose initial upward speed was 100 feet per second?
The strength of a cylindrical rod depends on its diameter. The greater the diameter of the rod, the more weight it can support before collapsing. The table shows the maximum weight supported by a rod as a function of its diameter. How much weight can a 1.2-centimeter rod support before collapsing?