Rate Problems: The Rate Equation

Randy's bar serves customers every weeknight from 5:00 p.m. to 1:00 a.m.. At Randy's bar, Silvermoon beer is sold at a constant rate except during happy hour, from 6:30 p.m. to 7:30 p.m., when the rate triples. If a keg of Silvermoon beer is half empty at 5:00 p.m. on Monday, and two-thirds empty at the beginning of happy hour, when will it be completely empty?

Incorrect. [[Snippet]]

Incorrect. The keg was half empty (or half full, depending on how you look at it) and then emptied to one-third of the entire volume. [[Snippet]]

Incorrect. [[snippet]]

Incorrect. You didn't take the rate change (during happy hour) into account. [[snippet]]

According to the problem, during the happy hour, the rate is tripled: $$3 \times 10 = 30$$. | | Rate | Time | Work | |-------------------|-------------------------|-----------------------|--------| | 5:00 to 6:30 p.m. | $$\frac{15}{1.5} = 10$$ | $$1.5$$ | $$15$$ | | Happy hour | $$30$$ | $$\color{red}{\frac{30}{30} = 1}$$ | $$30$$ | So we started happy hour with 30 liters and emptied them all during that hour. Thus, the keg is empty at exactly 7:30 p.m.

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Very good. [[snippet]] Let's see how this question is solved when we __Plug In__ a good number such as 90. From 5:00 to 6:30 p.m., the keg is emptied from half empty ($$\frac{90}{2}=45$$ liters) to two-thirds empty, which means one-third full ($$\frac{90}{3}=30$$ liters). That means that 15 liters are consumed. A Work Table is shown below. | | Rate | Time | Work | |-------------------|-------------------------|-----------------------|--------| | 5:00 to 6:30 p.m. | $$\color{red}{\frac{15}{1.5} = 10}$$ | $$1.5$$ | $$15$$ |

6:45 p.m.

7:00 p.m.

7:30 p.m.

9:00 p.m.

9:30 p.m.

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