Section 1.9 Learning Objectives
1.9: Applications – Variation
- Solve a direct variation problem
- Solve an inverse variation problem
- Solve a joint variation problem
- Solve application problems involving direct, inverse, and joint variation
Variation
So many cars, so many tires.
Direct Variation
Variation equations are examples of rational formulas and are used to describe the relationship between variables. For example, imagine a parking lot filled with cars. The total number of tires in the parking lot is dependent on the total number of cars. Algebraically, you can represent this relationship with an equation.
[latex]\text{number of tires}=4\cdot\text{number of cars}[/latex]
The number 4 tells you the rate at which cars and tires are related. You call the rate the constant of variation. It’s a constant because this number does not change. Because the number of cars and the number of tires are linked by a constant, changes in the number of cars cause the number of tires to change in a proportional, steady way. This is an example of direct variation, where the number of tires varies directly with the number of cars.
You can use the car and tire equation as the basis for writing a general algebraic equation that will work for all examples of direct variation. In the example, the number of tires is the output, 4 is the constant, and the number of cars is the input. Let’s enter those generic terms into the equation. You get [latex]y=kx[/latex]. That’s the formula for all direct variation equations.
[latex]\text{number of tires}=4\cdot\text{number of cars}\\\text{output}=\text{constant}\cdot\text{input}[/latex]
Direct variation Equation
To describe the relationship of direct variation, or directly proportional, “y varies directly as x” translates to
[latex]y=kx[/latex],
where [latex]k[/latex] is referred to as the constant of variation.
Example 1
Solve for [latex]k[/latex], the constant of variation, if [latex]y[/latex] varies directly as [latex]x[/latex], where [latex]y=300[/latex] and [latex]x=10[/latex].
In the video that follows, we present an example of solving a direct variation equation where we first need to find k, and then use k to determine y when x changes to a new value.
Inverse Variation
Another kind of variation is called inverse variation. In these equations, the output equals a constant divided by the input variable that is changing. In symbolic form, this is the equation [latex] y=\frac{k}{x}[/latex].
Inverse variation Equation
To describe the relationship of inverse variation, or inversely proportional, “y varies inversely as x” translates to
[latex] y=\frac{k}{x}[/latex],
where [latex]k[/latex] is referred to as the constant of variation.
Note: Inverse variation relationship are sometimes also referred to as indirect varation. Therefore, a problem may read “y varies indirectly with x” instead of “y varies inversely as x.”
Example 2
Solve for k, the constant of variation, if y varies inversely as x, where [latex]x=5[/latex] and [latex]y=25[/latex].
In the video that follows, we present an example of solving an inverse variation equation where we first need to find k, and then use k to determine y when x changes to a new value.
One example of an inverse variation is the speed required to travel between two cities in a given amount of time.
Example 3
Let’s say you need to drive from Boston to Chicago, which is about 1,000 miles. The more time you have, the slower you can go. If you want to get there in 20 hours, you need to go 50 miles per hour (assuming you don’t stop driving!), because [latex] \frac{1,000}{20}=50[/latex]. But if you can take 40 hours to get there, you only have to average 25 miles per hour, since [latex] \frac{1,000}{40}=25[/latex].
The equation for figuring out how fast to travel from the amount of time you have is [latex] speed=\frac{miles}{time}[/latex]. This equation should remind you of the distance formula [latex] d=rt[/latex]. If you solve [latex] d=rt[/latex] for r, you get [latex] r=\frac{d}{t}[/latex], or [latex] speed=\frac{miles}{time}[/latex].
In the case of the Boston to Chicago trip, you can write [latex] s=\frac{1,000}{t}[/latex]. Notice that this is the same form as the inverse variation function formula, [latex] y=\frac{k}{x}[/latex].
In the next example, we will find the water temperature in the ocean at a depth of 500 meters. Water temperature is inversely proportional to depth in the ocean.
Water temperature in the ocean varies inversely with depth.
Example 4
The water temperature in the ocean varies inversely with the depth of the water. The deeper a person dives, the colder the water becomes. At a depth of 1,000 meters, the water temperature is 5º Celsius. What is the water temperature at a depth of 500 meters?
In the video that follows, we present an example of inverse variation.
Joint Variation
A third type of variation is called joint variation. Joint variation is the same as direct variation except there are two or more quantities. With these problems, as with the other variation problems, we first must solve for [latex]k[/latex], the constant of variation, and then use that to answer any other parts that the question asked.
Joint Variation Equation
To describe the relationship of joint variation, “y varies jointly with x and z translates to
[latex]y=kxz[/latex],
where [latex]k[/latex] is referred to as the constant of variation.
One example of joint variation can be found in the area of a rectangle. The area of a rectangle can be found using the formula [latex]A=lw[/latex], where [latex]l[/latex] is the length of the rectangle and [latex]w[/latex] is the width of the rectangle. If you change the width of the rectangle, then the area changes and similarly if you change the length of the rectangle then the area will also change. You can say that the area of the rectangle “varies jointly with the length and the width of the rectangle.”
Let’s look at another joint variation example, with the area of a triangle.
Example 5
The area of a triangle varies jointly with the lengths of its base and height. If the area of a triangle is 30 inches[latex]^{2}[/latex] when the base is 10 inches and the height is 6 inches, find the variation constant and the area of a triangle whose base is 15 inches and height is 20 inches.
In the example above, finding k to be [latex] \frac{1}{2}[/latex] shouldn’t be surprising. You know that the area of a triangle is one-half base times height, [latex] A=\frac{1}{2}bh[/latex]. The [latex] \frac{1}{2}[/latex] in this formula is exactly the same [latex] \frac{1}{2}[/latex] that you calculated in this example!
Another example of joint variation is within the formula for the volume of a cylinder, [latex]V=\pi {{r}^{2}}h[/latex]. The volume of the cylinder varies jointly with the square of the radius and the height of the cylinder. The constant of variation is [latex] \pi [/latex].
In the following video, we show an example of finding the constant of variation for another jointly varying relation.
Direct, Joint, and Inverse Variation
k is the constant of variation. In all cases, [latex]k\neq0[/latex].
- Direct variation: [latex]y=kx[/latex]
- Inverse variation: [latex] y=\frac{k}{x}[/latex]
- Joint variation: [latex]y=kxz[/latex]
Summary of Variation
Direct, inverse, and joint variation equations are examples of rational formulas. In direct variation, the variables have a direct relationship—as one quantity increases, the other quantity will also increase. As one quantity decreases, the other quantity decreases. In inverse variation, the variables have an inverse relationship—as one variable increases, the other variable decreases, and vice versa. Joint variation is the same as direct variation except there are two or more variables.