Max puts a stopper in the bathtub and turnsont he faucet at time \( t=0 \). After 5 minutes, there are 11 gallons of water in the tub and after 10 minutes, there are 22 gallons in the tub. Write an equation that relates the number of gallons, \( g \), to \( t \) minutes. Does the relationship represent a linear function? Explain.

To solve this problem, we need to establish a relationship between the number of gallons of water \( g \) in the bathtub and the time \( t \) in minutes. ### Step 1: Extract Known Conditions From the problem, we have the following data points: - At \( t = 5 \) minutes, \( g = 11 \) gallons. - At \( t = 10 \) minutes, \( g = 22 \) gallons. ### Step 2: Determine if the Relationship is Linear To check if the relationship is linear, we can calculate the rate of change (slope) between the two points. The slope \( m \) can be calculated using the formula: \[ m = \frac{g_2 - g_1}{t_2 - t_1} \] where: - \( (t_1, g_1) = (5, 11) \) - \( (t_2, g_2) = (10, 22) \) Substituting the values: \[ m = \frac{22 - 11}{10 - 5} = \frac{11}{5} = 2.2 \] Since the slope is constant, the relationship is linear. ### Step 3: Write the Equation The general form of a linear equation is: \[ g = mt + b \] where \( m \) is the slope and \( b \) is the y-intercept. We already found \( m = 2.2 \). To find \( b \), we can use one of the points. Let's use \( (5, 11) \): \[ 11 = 2.2(5) + b \] Calculating \( 2.2(5) \): \[ 11 = 11 + b \] This simplifies to: \[ b = 0 \] ### Final Equation Thus, the equation that relates the number of gallons \( g \) to time \( t \) in minutes is: \[ g = 2.2t \] ### Conclusion Yes, the relationship represents a linear function because the rate of change (slope) is constant, and we can express it in the form \( g = mt + b \) where \( m \) is the slope and \( b \) is the y-intercept.