To find the equation that models the amount of gas left in the car as Phan drives, we need to determine the relationship between the number of hours spent driving ([tex]\(h\)[/tex]) and the amount of gas left in the tank ([tex]\(g\)[/tex]).
We'll start by using the provided data points:
| Hours spent driving ([tex]\(h\)[/tex]) | 3 | 5 | 7 | 9 |
|-----------------------------|---|---|---|---|
| Amount of gas left ([tex]\(g\)[/tex]) | 12 | 8 | 4 | 0 |
We need to determine the slope ([tex]\(m\)[/tex]) and the y-intercept ([tex]\(b\)[/tex]) of the linear relationship [tex]\(g = mh + b\)[/tex].
1. Calculate the Slope ([tex]\(m\)[/tex]):
The slope of a line is determined using the formula:
[tex]\[
m = \frac{ \sum{(x_i - \overline{x})(y_i - \overline{y})} }{ \sum{(x_i - \overline{x})^2} }
\][/tex]
In this context, [tex]\(x\)[/tex] represents hours, [tex]\(h\)[/tex], and [tex]\(y\)[/tex] represents gallons of gas left, [tex]\(g\)[/tex].
- Mean of [tex]\(h\)[/tex]: [tex]\(\overline{h} = \frac{3 + 5 + 7 + 9}{4} = \frac{24}{4} = 6\)[/tex]
- Mean of [tex]\(g\)[/tex]: [tex]\(\overline{g} = \frac{12 + 8 + 4 + 0}{4} = \frac{24}{4} = 6\)[/tex]
- Calculate numerator:
[tex]\[
\sum{(h_i - \overline{h})(g_i - \overline{g})} = (3-6)(12-6) + (5-6)(8-6) + (7-6)(4-6) + (9-6)(0-6)
\][/tex]
[tex]\[
= (-3)(6) + (-1)(2) + (1)(-2) + (3)(-6)
\][/tex]
[tex]\[
= -18 - 2 - 2 - 18 = -40
\][/tex]
- Calculate denominator:
[tex]\[
\sum{(h_i - \overline{h})^2} = (3-6)^2 + (5-6)^2 + (7-6)^2 + (9-6)^2
\][/tex]
[tex]\[
= (-3)^2 + (-1)^2 + 1^2 + 3^2
\][/tex]
[tex]\[
= 9 + 1 + 1 + 9 = 20
\][/tex]
- Thus, the slope [tex]\(m\)[/tex] is:
[tex]\[
m = \frac{-40}{20} = -2.0
\][/tex]
2. Calculate the Y-intercept ([tex]\(b\)[/tex]):
The y-intercept can be found using the mean values and the slope:
[tex]\[
b = \overline{g} - m\overline{h} = 6 - (-2)(6) = 6 + 12 = 18
\][/tex]
3. Formulate the Equation:
The linear equation that represents the relationship between hours spent driving ([tex]\(h\)[/tex]) and the amount of gas left ([tex]\(g\)[/tex]) is:
[tex]\[
g = -2h + 18
\][/tex]
4. Determine the Total Amount of Gasoline Needed to Fill Her Tank:
The y-intercept ([tex]\(b = 18\)[/tex]) represents the initial amount of gasoline in Phan's tank before she starts driving.
Thus, the equation that models the amount of gas left in Phan's car as she drives is:
[tex]\[
g = 18 - 2h
\][/tex]
And it takes 18 gallons to fill her tank.
The correct answer to the question is:
[tex]\[
g=18-2h; \, 18 \text{ gallons}
\][/tex]
