Skills Review for Physical Applications and Moments and Centers of Mass

Learning Outcomes

  • Define direct variation and solve problems involving direct variation
  • Graph equations using a graphing utility

In the Physical Applications and Moments and Centers of Mass sections, we will use integration to solve real-life examples. To prepare for this, we will review how to write direct variation equations, determine where two functions intersect, and graph equations using a graphing utility.

Write Direct Variation Equations

A used-car company has just offered their best candidate, Nicole, a position in sales. The position offers 16% commission on her sales. Her earnings depend on the amount of her sales. For instance if she sells a vehicle for $4,600, she will earn $736. She wants to evaluate the offer, but she is not sure how. In this section we will look at relationships, such as this one, between earnings, sales, and commission rate.

In the example above, Nicole’s earnings can be found by multiplying her sales by her commission. The formula [latex]e = 0.16s[/latex] tells us her earnings, [latex]e[/latex], come from the product of 0.16, her commission, and the sale price of the vehicle, [latex]s[/latex]. If we create a table, we observe that as the sales price increases, the earnings increase as well, which should be intuitive.

[latex]s[/latex], sales prices [latex]e = 0.16s[/latex] Interpretation
$4,600 [latex]e=0.16(4,600)=736[/latex] A sale of a $4,600 vehicle results in $736 earnings.
$9,200 [latex]e=0.16(9,200)=1,472[/latex] A sale of a $9,200 vehicle results in $1472 earnings.
$18,400 [latex]e=0.16(18,400)=2,944[/latex] A sale of a $18,400 vehicle results in $2944 earnings.

Notice that earnings are a multiple of sales. As sales increase, earnings increase in a predictable way. Double the sales of the vehicle from $4,600 to $9,200, and we double the earnings from $736 to $1,472. As the input increases, the output increases as a multiple of the input. A relationship in which one quantity is a constant multiplied by another quantity is called direct variation. Each variable in this type of relationship varies directly with the other.

The graph below represents the data for Nicole’s potential earnings. We say that earnings vary directly with the sales price of the car. The formula [latex]y=k{x}^{n}[/latex] is used for direct variation. The value [latex]k[/latex] is a nonzero constant greater than zero and is called the constant of variation. In this case, [latex]k=0.16[/latex] and [latex]n=1[/latex].

Graph of y=(0.16)x where the horizontal axis is labeled,

A General Note: Direct Variation

If [latex]x[/latex] and [latex]y[/latex] are related by an equation of the form

[latex]y=k{x}^{n}[/latex]

then we say that the relationship is direct variation and [latex]y[/latex] varies directly with the [latex]n[/latex]th power of [latex]x[/latex]. In direct variation relationships, there is a nonzero constant ratio [latex]k=\dfrac{y}{{x}^{n}}[/latex], where [latex]k[/latex] is called the constant of variation, which help defines the relationship between the variables.

How To: Given a description of a direct variation problem, solve for an unknown

  1. Identify the input, [latex]x[/latex], and the output, [latex]y[/latex].
  2. Determine the constant of variation. You may need to divide [latex]y[/latex] by the specified power of [latex]x[/latex] to determine the constant of variation.
  3. Use the constant of variation to write an equation for the relationship.
  4. Substitute known values into the equation to find the unknown.

Example: Writing a Direct Variation Equation

The quantity [latex]y[/latex] varies directly with the cube of [latex]x[/latex]. If [latex]y=25[/latex] when [latex]x=2[/latex], write the equation that represents this relationship. Then, find [latex]y[/latex] when [latex]x[/latex] is 6.

Try It

The quantity [latex]y[/latex] varies directly with the square of [latex]y[/latex]. If [latex]y=24[/latex] when [latex]x=3[/latex], find [latex]y[/latex] when [latex]x[/latex] is 4.

Try It

Watch this video to see a quick lesson about direct variation. You will see more worked examples.

You can view the transcript for “Direct Variation Applications” here (opens in new window).

Determine Where Two Functions Intersect

(see Module 6, Skills Review for Areas Between Curves, Determining Volume by Slicing, and Volumes of Revolution: Cylindrical Shells.)

Graph Equations Using a Graphing Utility

Please watch the video below to see how the Desmos Online Graphing Calculator can be used to graph equations. Although the video focuses specifically on graphing lines, note that any equation can be graphed with a graphing utility. There are also many different graphing utilities (or an actual physical graphing calculator) to use, but we love Desmos because of its excellent features, accessibility, user interface, and user experience.

You can view the transcript for “Learn Desmos: Lines” here (opens in new window).