https://file.aiquickdraw.com/mathbot/file/1727364604332hxg8df89.jpg

Let's go over the problem from the image.

The graph shows the number of gallons of gas remaining in the tank (\( G \)) as a function of the number of hours driven. The task is to:

1. Write the equation of the function \( G \).
2. Determine the \( x \)-intercept and explain its meaning in the context of the problem.

### Step 1: Identify key points from the graph
Looking at the graph, it seems to represent a linear relationship between the number of gallons remaining and the number of hours driven.

We can identify two key points from the graph (approximating if needed):
- When \( \text{hours} = 0 \), \( G = 15 \) gallons (this is the y-intercept).
- When \( \text{hours} = 5 \), \( G = 0 \) gallons (this is the x-intercept).

### Step 2: Find the slope of the line
The slope \( m \) of a line is given by the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using the two points \( (0, 15) \) and \( (5, 0) \):
\[
m = \frac{0 - 15}{5 - 0} = \frac{-15}{5} = -3
\]
So the slope of the line is \( -3 \).

### Step 3: Write the equation of the line
The equation of a line in slope-intercept form is:
\[
y = mx + b
\]
where \( m \) is the slope and \( b \) is the y-intercept.

From the graph, the y-intercept (\( b \)) is 15, and we just found the slope \( m = -3 \). So the equation is:
\[
G = -3t + 15
\]
where \( t \) is the number of hours driven and \( G \) is the gallons of gas remaining.

### Step 4: Determine the x-intercept
The x-intercept occurs when \( G = 0 \):
\[
0 = -3t + 15
\]
Solving for \( t \):
\[
3t = 15 \implies t = 5
\]
So the x-intercept is \( t = 5 \), which means after 5 hours of driving, the gas tank is empty.

### Interpretation of the x-intercept:
The x-intercept represents the total number of hours Tristan can drive before the gas runs out, which is 5 hours.

---

Would you like further clarification or more details on any step?

Here are 5 related questions for practice:

1. How would the equation change if the gas tank initially had 20 gallons?
2. What would happen to the slope if Tristan's car consumed gas more slowly?
3. How can you determine the average rate of gas consumption from the graph?
4. What does the slope of the line physically represent in this context?
5. How would you determine the total amount of gas used after driving for 2 hours?

**Tip:** Always look at the slope as the rate of change—here, it represents how quickly the gas is being used!

Tristan filled up his car with gas before embarking on a road trip across the country. Let G represent the number of gallons of gas remaining in his gas tank after driving for t hours. A graph of G is shown below. Write an equation for G, then state the x-intercept of the graph and determine its interpretation in the context of the problem.

Learn how to write a linear equation representing gas consumption and interpret the x-intercept using graph data. The problem involves understanding slope and intercepts, suitable for Grade 8-10 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation