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Titus works at a hotel. Part of his job is to keep the complimentary pitcher of water at least half full and always with ice. When he starts his shift, the water level shows 8 gallons, or 128 cups of water. As the shift progresses, he records the level of the water every 10 minutes. After 2 hours, he uses a regression calculator to compute an equation for the decrease in water. His equation is [tex]W \approx -0.414 t + 129.549[/tex], where [tex]t[/tex] is the number of minutes and [tex]W[/tex] is the level of water.

According to the equation, after about how many minutes would the water level be less than or equal to 64 cups?

A. 150 minutes
B. 160 minutes
C. 170 minutes
D. 180 minutes



Answer :

GinnyAnswer
To solve the problem, we need to determine the time [tex]\( t \)[/tex] when the water level [tex]\( W \)[/tex] becomes less than or equal to 64 cups, based on the given equation:

[tex]\[ W = -0.414t + 129.549 \][/tex]

Here's a step-by-step breakdown:

1. Set Up the Inequality: We need the water level [tex]\( W \)[/tex] to be less than or equal to 64 cups.
[tex]\[ -0.414t + 129.549 \leq 64 \][/tex]

2. Isolate the Term with [tex]\( t \)[/tex]: Subtract 129.549 from both sides of the inequality to isolate the term that includes [tex]\( t \)[/tex].
[tex]\[ -0.414t \leq 64 - 129.549 \][/tex]
[tex]\[ -0.414t \leq -65.549 \][/tex]

3. Solve for [tex]\( t \)[/tex]: Divide both sides of the inequality by -0.414. Note that since we are dividing by a negative number, we need to reverse the inequality sign.
[tex]\[ t \geq \frac{-65.549}{-0.414} \][/tex]

4. Calculate the Division:
[tex]\[ t \geq 158.34 \][/tex]

5. Approximate the Time: We need to determine the closest option for the number of minutes from the given choices. The closest number to 158.34 minutes is 160 minutes.

6. Conclusion: Based on the calculated time and the choices provided, the water level would be less than or equal to 64 cups after approximately 160 minutes.

Therefore, the answer is:
[tex]\[ \boxed{160 \text{ minutes}} \][/tex]