11. your uncle just bought a hybrid car and wants to take you and your siblings camping. the ratios represent the number of gallons remaining to hours of driving. create a table then graph and explain if the quantities are proportional to each other or not.
8 to 0
after 1 hour of driving, there are 6 gallons of gas left in the tank.
4 : 4
2 to 7
0/8
graph
proportional or not? explain.

Question

Accepted Answer

### Step 1: Organize the Data into a Table
We have the following data points:
- When hours of driving ($x$) = 0, gallons remaining ($y$) = 8 (from "8 to 0")
- When $x = 1$, $y = 6$ (given: "After 1 hour of driving, there are 6 gallons...")
- When $x = 4$, $y = 4$ (from "4 : 4")
- When $x = 7$, $y = 2$ (from "2 to 7")
- When $x = 8$, $y = 0$ (from "0/8")

So the table is:

| Hours of Driving ($x$) | Gallons Remaining ($y$) |
|--------------------------|---------------------------|
| 0                        | 8                         |
| 1                        | 6                         |
| 4                        | 4                         |
| 7                        | 2                         |
| 8                        | 0                         |

### Step 2: Analyze Proportionality
For two quantities to be proportional, the ratio $\frac{y}{x}$ should be constant for all $x
eq0$, and the graph should pass through the origin $(0,0)$.

- For $x = 1$, $\frac{y}{x}=\frac{6}{1}=6$
- For $x = 4$, $\frac{y}{x}=\frac{4}{4}=1$
- For $x = 7$, $\frac{y}{x}=\frac{2}{7}\approx0.286$

The ratios $\frac{y}{x}$ are not constant. Also, when $x = 0$, $y = 8$ (not $0$), so the graph does not pass through the origin.

### Step 3: Graph the Data
To graph the data:
- Plot the points $(0, 8)$, $(1, 6)$, $(4, 4)$, $(7, 2)$, $(8, 0)$ on a coordinate plane with the x - axis as "Hours of Driving" and the y - axis as "Gallons Remaining".
- Connect the points. The line will be a straight line (since the rate of change is constant: the slope $m=\frac{\Delta y}{\Delta x}$. From $x = 0,y = 8$ to $x = 1,y = 6$, $\Delta y=6 - 8=- 2$, $\Delta x=1 - 0 = 1$, slope $m=-2$. Let's check between $x = 1,y = 6$ and $x = 4,y = 4$: $\Delta y=4 - 6=-2$, $\Delta x=4 - 1 = 3$, slope $m=\frac{-2}{3}
eq - 2$? Wait, no—wait, let's recalculate the slope between consecutive points:

From $(0,8)$ to $(1,6)$: slope $m_1=\frac{6 - 8}{1 - 0}=\frac{-2}{1}=-2$

From $(1,6)$ to $(4,4)$: slope $m_2=\frac{4 - 6}{4 - 1}=\frac{-2}{3}\approx - 0.667$

Wait, actually, let's check the linearity. Let's see the change in $y$ with respect to $x$:

From $x = 0$ to $x = 1$: $y$ decreases by 2 (from 8 to 6)

From $x = 1$ to $x = 4$: $x$ increases by 3, $y$ decreases by 2 (from 6 to 4)

From $x = 4$ to $x = 7$: $x$ increases by 3, $y$ decreases by 2 (from 4 to 2)

From $x = 7$ to $x = 8$: $x$ increases by 1, $y$ decreases by 2 (from 2 to 0)

Wait, actually, the rate of decrease is not constant in terms of "per hour", but the relationship is linear. Let's find the equation of the line. Using two points, say $(0,8)$ and $(8,0)$. The slope $m=\frac{0 - 8}{8 - 0}=\frac{-8}{8}=-1$. Wait, but when $x = 1$, using $y=mx + b$ with $m=-1$ and $b = 8$ (since when $x = 0$, $y = 8$), $y=-x + 8$. Let's check $x = 1$: $y=-1 + 8 = 7$, but the given value is $y = 6$. Wait, there's a mistake in my earlier slope calculation. Let's use the points $(0,8)$, $(1,6)$, $(4,4)$, $(7,2)$, $(8,0)$. Let's calculate the slope between $(0,8)$ and $(8,0)$: $m=\frac{0 - 8}{8 - 0}=-1$. Between $(1,6)$ and $(8,0)$: $m=\frac{0 - 6}{8 - 1}=\frac{-6}{7}\approx - 0.857$. Between $(4,4)$ and $(8,0)$: $m=\frac{0 - 4}{8 - 4}=\frac{-4}{4}=-1$. Between $(7,2)$ and $(8,0)$: $m=\frac{0 - 2}{8 - 7}=-2$. So the points do not lie on a single straight line? Wait, no—wait the points $(0,8)$, $(1,6)$, $(4,4)$, $(7,2)$, $(8,0)$: Let's check the differences in $x$ and $y$:

From $x = 0$ to $x = 1$: $\Delta x = 1$, $\Delta y=-2$

From $x = 1$ to $x = 4$: $\Delta x = 3$, $\Delta y=-2$

From $x = 4$ to $x = 7$: $\Delta x = 3$, $\Delta y=-2$

From $x = 7$ to $x = 8$: $\Delta x = 1$, $\Delta y=-2$

Ah, so the change in $y$ is $-2$ for every $\Delta x$ of 1 (when $\Delta x = 1$) or 3 (when $\Delta x = 3$). Wait, actually, the slope between $(0,8)$ and $(1,6)$ is $-2$ (per hour), between $(1,6)$ and $(4,4)$ is $\frac{-2}{3}$ (per hour), which is not the same. But the key for proportionality is the constant ratio $\frac{y}{x}$ and passing through the origin. Since when $x = 0$, $y = 8$ (not 0), and the ratio $\frac{y}{x}$ is not constant, the quantities are not proportional.

### Step 4: Graph Description
To graph the data:
- Label the x - axis as "Hours of Driving" and the y - axis as "Gallons Remaining".
- Plot the points $(0,8)$, $(1,6)$, $(4,4)$, $(7,2)$, $(8,0)$.
- Connect the points. The graph is a straight line? Wait, let's check the equation. Let's use two points, say $(0,8)$ and $(8,0)$. The equation of the line is $y=-x + 8$ (since slope $m=-1$, y - intercept $b = 8$). Now check if the other points lie on this line:

- For $x = 1$: $y=-1 + 8 = 7$, but the given $y = 6$. So there's a discrepancy. Wait, maybe the "ratios" are misinterpreted. The problem says "the ratios represent the number of gallons remaining to hours of driving". So "8 to 0" means gallons : hours = 8:0, "4:4" means 4:4, etc. So maybe the data is:

- 8 gallons remaining when 0 hours driven (x=0, y=8)
- 6 gallons remaining when 1 hour driven (x=1, y=6)
- 4 gallons remaining when 4 hours driven (x=4, y=4)
- 2 gallons remaining when 7 hours driven (x=7, y=2)
- 0 gallons remaining when 8 hours driven (x=8, y=0)

Now, let's check the slope between (0,8) and (8,0): $m=\frac{0 - 8}{8 - 0}=-1$. So the equation is $y=-x + 8$. Now, check (1,6): $y=-1 + 8 = 7
eq6$. (4,4): $y=-4 + 8 = 4$ (matches). (7,2): $y=-7 + 8 = 1
eq2$. Wait, (4,4) matches, (8,0) matches, (0,8) matches, but (1,6) and (7,2) do not. So maybe there's a typo, but based on the given data, the key is proportionality.

### Final Answer for Proportionality
The quantities (hours of driving and gallons remaining) are not proportional.

### Explanation for Proportionality
1. **Constant Ratio Check**: For proportional quantities, $\frac{\text{gallons remaining}}{\text{hours of driving}}$ should be constant. For $x = 1$, $\frac{6}{1}=6$; for $x = 4$, $\frac{4}{4}=1$; for $x = 7$, $\frac{2}{7}\approx0.286$. These ratios are not constant.
2. **Origin Check**: Proportional relationships pass through the origin $(0,0)$. Here, when hours of driving ($x = 0$), gallons remaining ($y = 8$), not $0$, so the graph does not pass through the origin.

### Graph Description
- **Axes**: Label the x - axis "Hours of Driving" and the y - axis "Gallons Remaining".
- **Plot Points**: Mark the points $(0,8)$, $(1,6)$, $(4,4)$, $(7,2)$, $(8,0)$ on the coordinate plane.
- **Connect Points**: Draw a line (or line segments) connecting these points. The graph is a straight line? Wait, between $(0,8)$, $(4,4)$, $(8,0)$, the slope is $-1$, but $(1,6)$ and $(7,2)$ deviate. However, visually, it's a line that starts at $(0,8)$ and ends at $(8,0)$, passing through $(4,4)$, with $(1,6)$ and $(7,2)$ also on or near it (maybe a rounding or mis - ratio interpretation). But the key for proportionality is still the non - constant ratio and non - origin passage.

# Answer (Proportional or not? Explain.)
The quantities (hours of driving and gallons remaining) are not proportional. Explanation: For two quantities to be proportional, the ratio of gallons remaining to hours of driving ($\frac{y}{x}$) must be constant for all $x
eq0$, and the graph must pass through the origin $(0,0)$. Here, the ratio $\frac{y}{x}$ is not constant (e.g., $\frac{6}{1}=6$, $\frac{4}{4}=1$, $\frac{2}{7}\approx0.286$) and when $x = 0$ (0 hours driven), $y = 8$ (not 0), so the graph does not pass through the origin. The table and graph confirm these non - proportional characteristics.

(For the graph, plot the points $(0,8)$, $(1,6)$, $(4,4)$, $(7,2)$, $(8,0)$ on a coordinate plane with appropriate axis labels and connect them.)

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QUESTION IMAGE

Question

Explanation:

Step 1: Organize the Data into a Table

Step 2: Analyze Proportionality

Step 3: Graph the Data

Graph Description

Answer: