To find the linear equation that represents the relationship between the number of days collected (x) and the number of cans donated (y), we can use the data from the table:
- When \( x = 4 \), \( y = 955 \)
- When \( x = 7 \), \( y = 1660 \)
- When \( x = 10 \), \( y = 2365 \)
First, we will calculate the slope (m) of the line using two of the points. The formula for slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Let's choose the points \( (4, 955) \) and \( (7, 1660) \):
\[ m = \frac{1660 - 955}{7 - 4} = \frac{705}{3} = 235 \]
Now that we have the slope \( m = 235 \), we can use the point-slope form of a line \( y - y_1 = m(x - x_1) \) to find the linear equation. We will use the point \( (4, 955) \):
\[ y - 955 = 235(x - 4) \]
Expanding this:
\[ y - 955 = 235x - 940 \]
Adding 955 to both sides:
\[ y = 235x + 15 \]
The linear equation that represents the scenario is:
\[ \boxed{y = 235x + 15} \]
Thus, from the options provided, the correct equation is:
\[ y = 235x + 15 \]