Question

Katie noticed that after 1 hour of driving, she had 6.75 gallons of gas. After 6 hours of driving, there was only 1.5 gallons left. (Hint: Leave your answers as decimals). A.) Represent the information as two coordinate pairs in the form of (h, g), where h is the number of hours driven and g is the gallon of gas left. B.) Calculate the slope between the two coordinates. C.) Assuming the relationship between h and g is linear, create an equation for g in terms of h. D.) How much gas would she have left after dving for 3 hours?

Answer 1

We know that after 1 hour of driving, Katie had 6.75 gallons of gas. Also, after 6 hours of driving, she had 1.5 gallons left.

Part A

We can write these values as coordinate points, on the form:

Thus, the phrase: "after 1 hour of driving, Katie had 6.75 gallons of gas." can be represented as the point:

$(1,6.75)$

And: "After 6 hours of driving, there was only 1.5 gallons left" will have the point:

$(6,1.5)$

Part B

With this in mind, we can find the slope of a linear function that passes through those two points. Then,

$m=\frac{y_2-y_1}{x_2-x_1}=\frac{1.5-6.75}{6-1}=\frac{-5.25}{5}=-1.05$

Thus, the slope is -1.05.

Part C

Now, we will write a linear equation that relates g and h. As we already have the slope, we can use it to find the y-intercept as:

$\begin{gathered} y=mx+b \\ g=mh+b\text{ Writing it with the variables g and h.} \\ 1.5=(-1.05)(6)+b \\ 1.5+6.3=b \\ 7.8=b \end{gathered}$

This means that an equation for g in terms of h is given by:

$g=-1.05h+7.8$

Part D

For this part, we will use the linear equation we obtained for getting how gas she will have left after driving for 3 hours. As such, the value of h will be 3, and we replace on the function to obtain:

$\begin{gathered} g=-1.05(3)+7.8 \\ =-3.15+7.8 \\ =4.65 \end{gathered}$

This means that Katie will have 4.65 gallons left after driving for 3 hours.