To find the equation of the linear function representing the total amount of water \( y \) in the bathtub as a function of time \( r \), we can start by determining the slope of the line that passes through the points (4, 24) and (6, 30).
The formula for the slope \( m \) between two points \((r_1, y_1)\) and \((r_2, y_2)\) is given by:
\[ m = \frac{y_2 - y_1}{r_2 - r_1} \]
Plugging in the coordinates of the given points:
\[ m = \frac{30 - 24}{6 - 4} = \frac{6}{2} = 3 \]
Now that we have the slope \( m = 3 \), we can use the point-slope form of a linear equation:
\[ y - y_1 = m(r - r_1) \]
Using the point (4, 24):
\[ y - 24 = 3(r - 4) \]
Expanding this equation:
\[ y - 24 = 3r - 12 \] \[ y = 3r + 12 \]
So the equation of the function is:
\[ y = 3r + 12 \]
Next, to find out how much water was in the bathtub when Mia turned on the faucet (i.e., when \( r = 0 \)), we substitute \( r = 0 \) into the equation:
\[ y = 3(0) + 12 = 12 \]
Therefore, the amount of water in the bathtub when Mia turns on the faucet is 12 gallons.