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Rates of Change

Understand slope as rate of change

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Rates of Change
Suppose a new gym opened, and it had 20 members after 1 week, 40 members after 2 weeks, and 60 members after 3 weeks. Could you calculate the rate of change in the number of gym members? How is this different than the slope? If this rate continues, how long will it take for the gym to have 300 members?

Finding the Rate of Change

When finding the slope of real-world situations, it is often referred to as rate of change. “Rate of change” means the same as “slope.” If you are asked to find the rate of change, use the slope formula or make a slope triangle.

Let's use rate of change to solve the following problems:

  1. Andrea has a part-time job at the local grocery store. She saves for her vacation at a rate of $15 every week. What is the rate of change in the amount of money Andrea has?

Begin by finding two ordered pairs. You can make a chart or use the Substitution Property to find two coordinates.

Using the points (2, 30) and (10, 150). Since (2, 30) is written first, it can be considered the first point, \begin{align*}(x_1, \ y_1).\end{align*} That means \begin{align*}(10,150)=(x_2, \ y_2).\end{align*} (Note that it doesn't matter at all which point is considered first, as the slope will end up the same either way.)

Use the formula: \begin{align*}slope=\frac{y_2-y_1}{x_2-x_1}=\frac{150-30}{10-2}=\frac{120}{8}=\frac{15}{1}\end{align*}

Andrea’s rate of change is \begin{align*}\frac{$15}{\text{1 week}}\end{align*} 

  1. A candle has a starting length of 10 inches. Thirty minutes after lighting it, the length is 7 inches. Determine the rate of change in the length of the candle as it burns. How long does it take for the candle takes to completely burn to nothing?

Begin by finding two ordered pairs. The candle begins at 10 inches in length. So at time “zero”, the length is 10 inches. The ordered pair representing this is (0, 10). 30 minutes later, the candle is 7 inches, so the ordered pair is (30, 7). Since (0, 10) is written first, it can be called \begin{align*}(x_1, \ y_1).\end{align*} That means \begin{align*}(30,7)=(x_2, \ y_2).\end{align*}

Use the formula: \begin{align*}slope=\frac{y_2-y_1}{x_2-x_1}=\frac{7-10}{30-0}=\frac{\text{-}3}{30}=\text{-}\frac{1}{10}\end{align*}

The candle has a rate of change of -1 inch/10 minutes (the rate is negative because the candle is getting shorter over time). To find the length of time it will take for the candle to burn out, you can create a graph, use guess and check, or solve an equation.

You can create a graph to help visualize the situation. By plotting the ordered pairs you were given and by drawing a straight line connecting them, you can estimate it will take 100 minutes for the candle to burn out.

  1. Examine the following graph. It represents a journey made by a large delivery truck on a particular day. During the day, the truck made two deliveries, each one taking one hour. The driver also took a one-hour break for lunch. Identify what is happening at each stage of the journey (stages \begin{align*}A\end{align*} through \begin{align*}E\end{align*}).

Here is the driver's journey.

A. The truck sets off and travels 80 miles in 2 hours.

B. The truck covers no distance for 1 hour.

C. The truck covers \begin{align*}(120 - 80) = 40\end{align*} miles in 1 hour.

D. The truck covers no distance for 2 hours.

E. The truck covers 120 miles in 2 hours.

To identify what is happening at each leg of the driver’s journey, you are being asked to find each rate of change.

The rate of change for line segment \begin{align*}A\end{align*} can be found using either the formula or the slope triangle. Using the slope triangle, \begin{align*}\text{vertical change}=80\end{align*} and the \begin{align*}\text{horizontal change}=2.\end{align*}

\begin{align*}\text{slope}=\frac{\text{rise}}{\text{run}}=\frac{\text{80 miles}}{\text{2 hours}}= \frac{\text{40 miles}}{\text{1 hour}}\end{align*}.

Segments \begin{align*}B\end{align*} and \begin{align*}D\end{align*} are horizontal lines and each has a slope of zero.

The rate of change for line segment \begin{align*}C\end{align*} using the slope formula: \begin{align*}\text{Rate of change} = \frac{\triangle y}{\triangle x} = \frac{(120-80) \text{ miles}}{(4-3) \text{ hours}}= 40 \ \text{miles per hour}\end{align*}.

The rate of change for line segment \begin{align*}E\end{align*} using the slope formula: \begin{align*}\text{Rate of change} = \frac{\triangle y}{\triangle x} = \frac{(0-120) \text{ miles}}{(8-6) \text{ hours}}= \frac{\text{-}120 \text{ miles}}{2 \text{ hours}}= \text{-}60 \ \text{miles per hour}.\end{align*}

The truck is traveling at negative 60 mph. Another way to say this is that the truck is returning home at a rate of 60 mph.

   

 

 

Examples

Example 1

Earlier, you were told that a new gym had 20 members after 1 week, 40 members after 2 weeks, and 60 members after 3 weeks. What is the rate of change in the number of gym members? If this rate continues, how long will it take for the gym to have 300 members?

We have three points to use to calculate the rate of change: (1, 20), (2, 40), and (3, 60). Since only two points are needed to find the rate of change, you can pick any two points and use the slope formula. Note that this only works when the points represent a constant rate of change or a linear equation.

\begin{align*}slope=\frac{y_2-y_1}{x_2-x_1}=\frac{40-20}{2-1}=\frac{20}{1}=20\end{align*}.

Therefore the rate of change is 20 members per week. If you choose two other points out of the three, you will get the same result. 

Two determine how long it will take for the gym to have 300 members, you can use proportions:

\begin{align*}\frac{20 \text{ members}}{1 \text{ week}} = \frac{ 300 \text{ members}}{x\text{ weeks}}\end{align*} 

Solve for \begin{align*}x\end{align*} using cross multiplication:

\begin{align*}\frac{20}{1}=\frac{300}{x}\\ 20\times x = 300\times 1\\ 20x = 300\\ 20x\div 20 = 300 \div 20\\ x = 15\end{align*} 

It will take 15 weeks for the gym to have 300 members if it continues at a rate of adding 20 members per week.

Example 2

Adel spent $125 on groceries in one week. Use this to predict how much Adel will spend per month on groceries, if she keeps buying them at the same rate.

Adel's weekly rate is $125 per week. Since a month is about 4 weeks, multiply $125/week times 4 weeks:

\begin{align*} \frac{\$125}{1 \text{ week}} \cdot 4 \text{ weeks}= \frac{\$125}{1 \cancel{\text{ week}}} \cdot 4 \cancel{\text{ weeks}}=\$125\cdot 4=\$500\end{align*}

Adel will spend about $500 on groceries per month.

Review

  1. How is slope related to rate of change? In what ways is it different?
  1. The graph below is a distance-time graph for Mark’s 3.5-mile cycle ride to school. During this particular ride, he rode on cycle paths but the terrain was hilly. His speed varied depending upon the steepness of the hills. He stopped once at a traffic light and at one point he stopped to mend a tire puncture. Identify each section of the graph accordingly.

  1. Four hours after she left home, Sheila had traveled 145 miles. Three hours later she had traveled 300 miles. What was her rate of change?
  2. Jenna saves $60 every \begin{align*}2\frac{1}{2} \text{ weeks}.\end{align*} What is the rate of change of her savings?
  3. Geoffrey has a rate of change of \begin{align*}\frac{\text{10 feet}}{\text{1 second}}.\end{align*} Write a situation that could fit this slope.

Quick Quiz

  1. Find the intercepts of \begin{align*}3x+6y=25\end{align*} and graph the equation.
  2. Find the slope between (8, 5) and (–5, 6).
  3. Graph \begin{align*}f(x)=2x+1\end{align*}
  4. Graph the ordered pair with the following directions: 4 units west and 6 units north of the origin.
  5. Using the graph below, list two “trends” about this data. A trend is something you can conclude about the given data.

Review (Answers)

To see the Review answers, open this PDF file and look for section 4.8. 

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Vocabulary

rate of change

When finding the slope of real-world situations, it is often referred to as rate of change. Rate of change means the same as slope.

Horizontally

Horizontally means written across in rows.

Slope

Slope is a measure of the steepness of a line. A line can have positive, negative, zero (horizontal), or undefined (vertical) slope. The slope of a line can be found by calculating “rise over run” or “the change in the y over the change in the x.” The symbol for slope is m

Vertically

Vertically means written up and down in columns.

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