A.  First, we must be careful with the order we use to set up our rate.  Since we are told to compare miles to gallons, we will put the miles in the numerator and gallons in the denominator. To set up the rate, we simply use the given values.

[latex]Rate=\frac{216 \hspace{.05in} miles}{9 \hspace{.05in} gallons}[/latex], or equivalently [latex]216[/latex] miles per [latex]9[/latex] gallons

To transform this into a unit rate, we can simplify the fraction to lowest terms since 9 divides evenly into 216.

[latex]Unit \hspace{.05in} Rate=\frac{24 \hspace{.05in} miles}{1 \hspace{.05in} gallon}[/latex], or [latex]24 \hspace{.05in} miles \hspace{.05in} per \hspace{.05in} gallon[/latex]

B.  Now, the rate is [latex]\frac{223 \hspace{.05in} miles}{9 \hspace{.05in} gallons}[/latex]. However, you may notice that 223 is not divisible by 9. In this case, we just do the division, [latex]223 \div 9[/latex], potentially with the aid of a calculator. Round to the nearest tenth produces

[latex]Unit \hspace{.05in} Rate=24.8 \hspace{.05in}miles \hspace{.05in} per \hspace{.05in} gallon[/latex]

First, we need to create two ordered pairs from the data.

[latex](1998, 685)[/latex] and [latex](2000, 760)[/latex]

Notice that the “years” are the x-coordinates and the “number of jobs” are the y-coordinates.  That is so the number of jobs will be in the numerator in our slope formula.  Since we were asked to find the rate of change in jobs per year, we need the number of jobs to be in the numerator and the years to be in the denominator.

Next, we are going to use these two ordered pairs in the slope formula to find the average rate of change in jobs per year.

Rate of Change = Slope= [latex]\frac{y_2-y_1}{x_2-x_1}[/latex]

Rate of Change = Slope=[latex]\frac{760-685}{2000-1998}[/latex]

Rate of Change = Slope= [latex]\frac{75}{2}[/latex]

Rate of Change = Slope= 37.5 jobs/year

The number of living wage jobs increased by 37.5 jobs per year.

First, we need to create two ordered pairs from the data. It is important to label the variables correctly. When time is involved, time will usually be our x-variable. This is confirmed in that we want the rate of change in the population (so population on top) per year (therefore, years on bottom).

[latex](2012, 3267100)[/latex] and [latex](2002, 3289200)[/latex]

Use these two ordered pairs in the slope formula to find the average rate of change in people per year.

Rate of Change = Slope= [latex]\frac{y_2-y_1}{x_2-x_1}=\frac{\text{Change in population}}{\text{Change in time}}[/latex]

Rate of Change = Slope=[latex]\frac{3289200-3267100}{2002-2012}[/latex]

Rate of Change = Slope= [latex]\frac{22100}{-10}[/latex]

Rate of Change = Slope= [latex]\frac{2210\text{ people}}{1\text{ year}}[/latex]

The population decreased by an average of 2,210 people/year.

First, we need to create two ordered pairs from the data.  Again, it is important to label the variables correctly. When time is involved, time will usually be our x-variable. We will make two ordered pairs to represent (hours, miles).

[latex](3, 35)[/latex] and [latex](10, 112)[/latex]

Use these two ordered pairs in the slope formula to find the average rate in miles per hour during her trip.

Rate of Change = Slope= [latex]\frac{y_2-y_1}{x_2-x_1}=\frac{\text{Change in miles}}{\text{Change in time}}[/latex]

Rate of Change = Slope=[latex]\frac{112-35}{10-3}[/latex]

Rate of Change = Slope= [latex]\frac{77}{7}[/latex]

Rate of Change = Slope= [latex]\frac{11\text{ miles}}{1\text{ hour}}[/latex]

Michelle’s average rate was 11 miles per hour (mph).

Since [latex]x[/latex] represents the number of years since 1950, we will need to subtract 1950 from 2022 to determine what our  [latex]x[/latex]-value is.

[latex]2022-1950=72[/latex]

Next, we will substitute [latex]x=72[/latex] into the function.

[latex]f(x)=3966x+74,400[/latex]

[latex]f(72)=3966(72)+74,400[/latex]

[latex]f(72)=285,552+74,00[/latex]

[latex]f(72)=359,952[/latex]

Therefore, based on this linear model, we predict that the median value of a house in Hawaii in 2022 will be $359,952.

A.  The slope is [latex]m=-0.97[/latex].

B.  Since [latex]x[/latex] represents the annual tuition, we will substitute 30,000 for [latex]x[/latex]; or in other words, we are looking for [latex]f(30,000).

[latex]f(x)=-0.97x+159,000[/latex]

[latex]f(30,000)=-0.97(30,000)+159,000[/latex]

[latex]f(30,000)=-29,100+159,000[/latex]

[latex]f(30,000)=129,900[/latex]

Therefore, based on this model, a graduate of a college or university where the annual tuition is $30,000 can expect an average mid-career salary of $129,900.

C.  We already identified the slope to be [latex]m=-0.97[/latex], but what does this mean in the context of the problem? Recall from the first half of this section that slope is the rate of change, specifically the change in [latex]y[/latex] relative to changes in [latex]x[/latex]. In this problem, [latex]x[/latex] represents tuition while [latex]y[/latex] represent mid-career salary.

Based on the negative slope, we conclude that mid-career salary decreases as tuition increases.

A.  The slope is [latex]m=25[/latex].

B.  Since [latex]x=[/latex] the number of skyrockets produced, we will substitute [latex]x=20[/latex] into the function.

[latex]C(x)=25x+840[/latex]

[latex]C(20)=25(20)+840[/latex]

[latex]C(20)=500+840[/latex]

[latex]C(20)=1340[/latex]

Therefore, the cost to produce 20 skyrockets is $1,340.

C. In context, the slope here represent the average rate of change in cost relative to the change in rockets produced. So, the slope of [latex]m=25[/latex], or equivalently, [latex]m=\frac{25}{1}[/latex] indicates that the cost increases by $25 for each additional rocket produced.  Therefore, cost increases as the number of rockets produced increases.