To determine after how many minutes the water level would be less than or equal to 64 cups, we start with the given equation that models the decrease in water level over time:
[tex]\[ W \approx -0.414 t + 129.549 \][/tex]
Here, [tex]\( W \)[/tex] is the water level in cups, and [tex]\( t \)[/tex] is the number of minutes that have passed. We need to find the value of [tex]\( t \)[/tex] when [tex]\( W \leq 64 \)[/tex] cups.
Let's set [tex]\( W \)[/tex] to 64 and solve for [tex]\( t \)[/tex]:
[tex]\[ 64 = -0.414 t + 129.549 \][/tex]
To isolate [tex]\( t \)[/tex] on one side, we rearrange the equation as follows:
1. Subtract 64 from both sides:
[tex]\[ 129.549 - 64 = -0.414 t \][/tex]
2. Simplify the left side:
[tex]\[ 65.549 = -0.414 t \][/tex]
3. Divide both sides by -0.414 to solve for [tex]\( t \)[/tex]:
[tex]\[ t = \frac{65.549}{0.414} \][/tex]
By performing the division, we find:
[tex]\[ t \approx 158.33 \][/tex]
Now, we compare the calculated [tex]\( t \)[/tex] to the provided options: 150, 160, 170, and 180 minutes. The calculated time of approximately 158.33 minutes is closest to 160 minutes.
Thus, according to the equation, the water level would be less than or equal to 64 cups after about 160 minutes.
