Question

The relationship between Amanda’s distance from school and time since she left school is linear, as shown in the table below.
time (hours): 0 , 1, 3
distance (miles): 3, 4.25, 6.75
Based on the table, determine whether each statement is true or false.

Amanda's distance from school after 4 hours is 9.75 miles.

Answer 1

To determine whether the statement "Amanda's distance from school after 4 hours is 9.75 miles" is true or false, we first need to find the equation of the linear relationship between Amanda’s distance from school and time.

We can use the points given in the table: (0, 3), (1, 4.25), and (3, 6.75) to calculate the slope and then derive the equation of the line.

1. Calculate the slope (m) using two points. We'll use (0, 3) and (1, 4.25):

\[
m = \frac{(y_2 - y_1)}{(x_2 - x_1)} = \frac{(4.25 - 3)}{(1 - 0)} = \frac{1.25}{1} = 1.25
\]

2. Now we can use the point-slope form of the line to find the equation. Using the point (0, 3):

\[
y - y_1 = m(x - x_1)
\]
\[
y - 3 = 1.25(x - 0)
\]
\[
y = 1.25x + 3
\]

Now we have the equation of the line that represents Amanda's distance from school over time:

\[
y = 1.25x + 3
\]

3. We can now plug in \(x = 4\) (for 4 hours) into our equation to find Amanda's distance:

\[
y = 1.25(4) + 3
\]
\[
y = 5 + 3 = 8
\]

So Amanda's distance from school after 4 hours is 8 miles.

As a result, the statement "Amanda's distance from school after 4 hours is 9.75 miles" is **false**.