To analyze each statement based on the linear relationship defined in the table, we first need to determine the linear equation that relates time and distance.
From the provided data points:
- At time 0 hours, the distance from school is 3 miles.
- At time 1 hour, the distance from school is 4.25 miles.
- At time 3 hours, the distance from school is 6.75 miles.
1. Slope Calculation: Let's find the slope of the line, which represents the change in distance per hour. Using the data points (0, 3) and (1, 4.25):
\[ \text{slope} = \frac{\text{change in distance}}{\text{change in time}} = \frac{4.25 - 3}{1 - 0} = \frac{1.25}{1} = 1.25 \text{ miles/hour} \]
2. Equation of the Line: To find the equation of the line in the slope-intercept form (y = mx + b), we can use the slope we calculated and the initial condition (0, 3).
Using the point (0, 3):
- m (slope) = 1.25
- b (y-intercept) = 3
The equation is: \[ \text{Distance} = 1.25 \cdot \text{Time} + 3 \]
Now, let's evaluate each statement:
Statement 1: Amanda's initial distance from her school is 1.25 miles.
- False. The initial distance from school at time 0 is 3 miles, not 1.25 miles.
Statement 2: Amanda's distance increases by 1.25 miles each hour.
- True. The slope we calculated (1.25 miles/hour) indicates that Amanda's distance from the school increases by 1.25 miles for every hour that passes.
Statement 3: Amanda's distance from school after 4 hours is 9.75 miles.
- To find the distance after 4 hours, we can plug in 4 for time in our equation: \[ \text{Distance} = 1.25 \cdot 4 + 3 = 5 + 3 = 8 \text{ miles} \]
- False. After 4 hours, her distance from school is 8 miles, not 9.75 miles.
Summary:
- False
- True
- False
1 answer