Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

1.15.2 Modeling Equations Using Direct Variation

1.15.2 • Modeling Equations Using Direct Variation

Activity

In your group, work through your assigned problem. Then be ready to share.

Group 1. When Raoul runs on the treadmill at the gym, the number of calories, $c$ , he burns varies directly with the number of minutes, $m$ , he uses the treadmill. He burned 315 calories when he used the treadmill for 18 minutes. Write the equation that relates $c$ and $m$ .

Compare your answer:

$c = 17.5 m$

Group 2. The number of calories, $c$ , burned varies directly with the amount of time, $t$ , spent exercising. Arnold burned 312 calories in 65 minutes exercising. Write the equation that relates $c$ and $t$ .

Compare your answer:

$c = 4.8 m$

Group 3. The distance a moving body travels, $d$ , varies directly with time, $t$ , it moves. A train travels 100 miles in 2 hours. Write the equation that relates $d$ and $t$ .

Compare your answer:

$d = 50 t$

What is Direct Variation?

All of these are examples of direct variation.

Two variables vary directly if one is the product of a constant and the other.

For any two variables $x$ and $y$ , $y$ varies directly with $x$ if

$y = k x$ , where $k \neq 0$ .

The constant $k$ is called the constant of variation.

Look back at the problem you solved. Use the direct variation equation to solve it again using these steps:

Step 1 - Write the equation for direct variation.

Step 2 - Substitute the given values for the variables.

Step 3 - Solve for the constant of variation.

Step 4 - Write the equation that relates x and y using the constant of variation.

Let’s look at example 3 again and solve it with the direct variation equation.

The distance a moving body travels, $d$ , varies directly with time, $t$ , it moves. A train travels 100 miles in 2 hours. Write the equation that relates $d$ and $t$ .

Step 1 - Write the equation for direct variation.

$y = k x$

Step 2 - Substitute the given values for the variables.

$100 = k (2)$

Step 3 - Solve for the constant of variation.

$k = 50$

Step 4 - Write the equation that relates x and y using the constant of variation.

$y = 50 x$

Self Check

The price,

P

, that Eric pays for gas varies directly with the number of gallons,

g

, he buys. It costs him $65 to buy 25 gallons of gas. Which equation expresses the price of any amount of gas,

P

, in terms of the number of gallons,

g

?

$P=\frac{5}{13}g$
$P=\frac{65}{g}$
$P=25g$
$P=\frac{13}{5}g$

Additional Resources

Modeling Situations Using Direct Variation

When two quantities are related by a proportion, we say they are proportional to each other. Another way to express this relation is to talk about the variation of the two quantities. We will discuss direct variation here.

Lindsay gets paid $15 per hour at her job. If we let $s$ be her salary, in dollars per hour, and $h$ be the number of hours she has worked, we could model this situation with the equation

$s = 15 h$

Lindsay’s salary is the product of a constant, 15, and the number of hours she works. We say that Lindsay’s salary varies directly with the number of hours she works. Two variables vary directly if one is the product of a constant and the other.

Direct Variation

For any two variables $x$ and $y$ , $y$ varies directly with $x$ if

$y = k x$ , where $k \neq 0$

The constant $k$ is called the constant of variation.

In applications using direct variation, generally we will know values of one pair of the variables and will be asked to find the equation that relates $x$ and $y$ .

Here’s how to model an equation using direct variation for application problems.

Step 1 - Write the formula for direct variation.

Step 2 - Substitute the given values for the variables.

Step 3 - Solve for the constant of variation.

Step 4 - Write the equation that relates $x$ and $y$ using the constant of variation.

Let’s look at an application problem:

The cost of a pizza, $y$ , varies directly with its diameter, $x$ , and an 8-inch pizza costs $12. What is the equation that relates $x$ and $y$ ?

Step 1 - Write the formula for direct variation.

$y = k x$

Step 2 - Substitute the given values for the variables.

$12 = k (8)$

Step 3 - Solve for the constant of variation.

$k = \frac{12}{8} = \frac{3}{2}$

Step 4 - Write the equation that relates x and y using the constant of variation.

$y = \frac{3}{2} x$

Try it

Modeling Situations Using Direct Variation

The distance a moving body travels, $d$ , varies directly with the amount of time, $t$ , it moves. A train travels 60 miles in 2 hours. What is the equation that relates $d$ and $t$ ?

Compare your answer: Here is how to model this application problem using direct variation.

Step 1 - Write the formula for direct variation.

$y = k x$

$d = k t$

Step 2 - Identify the variables

$d$ : distance, $t$ : time

Step 3 - Substitute the given values for the variables.

$60 = k (2)$

Step 4 - Solve for the constant of variation, $k$ .

$k = 30$

Step 5 - Write the equation that relates d and $t$ using the constant of variation.

$d = 30 t$