To determine the correct statement, let's analyze the table and understand the linear function that models the cost to rent a movie.
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Nights rented, } x & \text{Total cost (dollars), } y \\
\hline
1 & 1.50 \\
\hline
2 & 2.75 \\
\hline
3 & 4.00 \\
\hline
4 & 5.25 \\
\hline
5 & 6.50 \\
\hline
\end{array}
\][/tex]
Given that the relationship is linear, the total cost \( y \) can be modeled by the equation of a line:
[tex]\[
y = mx + b
\][/tex]
where \( m \) is the slope (rate per night) and \( b \) is the y-intercept (initial cost when \( x = 0 \)).
To find the slope \( m \), we need to calculate the change in total cost (\( \Delta y \)) over the change in the number of nights rented (\( \Delta x \)):
First, calculate the differences between consecutive total costs (deltas):
[tex]\[
\begin{align*}
\Delta y_1 &= 2.75 - 1.50 = 1.25 \\
\Delta y_2 &= 4.00 - 2.75 = 1.25 \\
\Delta y_3 &= 5.25 - 4.00 = 1.25 \\
\Delta y_4 &= 6.50 - 5.25 = 1.25 \\
\end{align*}
\][/tex]
Similarly, calculate the differences between consecutive numbers of nights rented:
[tex]\[
\begin{align*}
\Delta x_1 &= 2 - 1 = 1 \\
\Delta x_2 &= 3 - 2 = 1 \\
\Delta x_3 &= 4 - 3 = 1 \\
\Delta x_4 &= 5 - 4 = 1 \\
\end{align*}
\][/tex]
Now, calculate the slope \( m \) by dividing \( \Delta y \) by \( \Delta x \):
[tex]\[
m = \frac{\Delta y}{\Delta x} = \frac{1.25}{1} = 1.25
\][/tex]
We obtained the same slope for each pair, indicating a constant rate per night and confirming the linear relationship.
So the slope \( m \) is 1.25 dollars per night.
This means:
- The rate per night for renting a movie is 1.25 dollars.
- The initial cost is given by the y-intercept \( b \), which we do not need to find for this problem since the question only pertains to the slope.
Hence, the correct statement is:
[tex]\[
\boxed{\text{D. The slope is 1.25. It represents the rate per night.}}
\][/tex]
