To find the linear equation that models the amount of water [tex]\( w \)[/tex] in the bottle in terms of the time [tex]\( t \)[/tex], we need to determine the slope (rate of water flow) and the y-intercept (initial amount of water in the bottle before we start timing).
1. Calculate the rate of water flow (slope, [tex]\( m \)[/tex]):
We are given the following data points:
- At [tex]\( t = 5 \)[/tex] seconds, [tex]\( w = 17 \)[/tex] ounces.
- At [tex]\( t = 11 \)[/tex] seconds, [tex]\( w = 35 \)[/tex] ounces.
The slope ([tex]\( m \)[/tex]) can be calculated using the formula for the rate of change:
[tex]\[
m = \frac{\Delta w}{\Delta t} = \frac{w_2 - w_1}{t_2 - t_1}
\][/tex]
Substituting the given values:
[tex]\[
m = \frac{35 - 17}{11 - 5} = \frac{18}{6} = 3
\][/tex]
So, the rate of water flow is [tex]\( 3 \)[/tex] ounces per second.
2. Determine the initial amount of water (y-intercept, [tex]\( b \)[/tex]):
To find the y-intercept [tex]\( b \)[/tex], we use one of the data points and the slope we just found. We use the point [tex]\( (t_1, w_1) = (5, 17) \)[/tex]:
[tex]\[
w = mt + b
\][/tex]
Plug in [tex]\( t = 5 \)[/tex], [tex]\( w = 17 \)[/tex], and [tex]\( m = 3 \)[/tex]:
[tex]\[
17 = 3(5) + b
\][/tex]
[tex]\[
17 = 15 + b
\][/tex]
Solving for [tex]\( b \)[/tex]:
[tex]\[
b = 17 - 15 = 2
\][/tex]
So, the initial amount of water in the bottle is [tex]\( 2 \)[/tex] ounces.
3. Form the linear equation:
Combining the slope ([tex]\( m \)[/tex]) and the y-intercept ([tex]\( b \)[/tex]), the linear equation that models the amount of water [tex]\( w \)[/tex] in the bottle in terms of time [tex]\( t \)[/tex] is:
[tex]\[
w = 3t + 2
\][/tex]
Therefore, the correct equation that models the amount of water in the bottle is:
[tex]\[
w = 3t + 2
\][/tex]
The correct answer is:
[tex]\[
w = 3t + 2
\][/tex]
