Filling Station Flow-Rate Claim
The environmental club at North Ridge High School is evaluating a vendor's claim that a new water bottle filling station dispenses water at 0.40 gallons per minute. Maya tests the station by filling a 1-liter bottle while timing it with a stopwatch that displays to the nearest 0.1 second. The stopwatch reads 38.4 seconds. The bottle's 1-liter mark is accurate to within +/-20 milliliters. The school enrolls 620 students, and the club estimates that 45% of them will refill a 20-fluid-ounce bottle once each school day. A 24-pack of 16.9-fluid-ounce bottled water costs $4.99. The club wants to know whether the vendor's flow-rate claim is supported by Maya's measurement and whether the station would eliminate at least 8,000 disposable bottles over a 180-day school year. They also want to report the flow rate with an appropriate level of precision, since the principal will use the number in a grant application.
1. Convert Maya's measurement to gallons per minute and determine whether the vendor's 0.40 gal/min claim is supported once the measurement uncertainty is considered. Show the conversion and the range of plausible rates.
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Use 1 L = 0.264 gallon, so 1 L/min would be 0.264 gal/min, but the time is 38.4 s = 0.64 min, not 1 min. The measured rate is 0.264 / 0.64 = 0.413 gal/min, about 0.41 gal/min. The uncertainty is not negligible: +/-20 mL = +/-0.02 L = +/-0.0053 gal, and +/-0.05 s = +/-0.00083 min. The smallest plausible rate is about (0.264 - 0.0053) / (0.64 + 0.00083) = 0.404 gal/min. The largest is about (0.264 + 0.0053) / (0.64 - 0.00083) = 0.422 gal/min. So the plausible range is roughly 0.404 to 0.422 gal/min. The vendor's 0.40 gal/min is just below that range. The claim is close but is not supported within the stated precision; the measured flow is about 0.41 gal/min, slightly faster than claimed.
2. Using a rate of 0.41 gal/min, evaluate the club's claim that the station will eliminate at least 8,000 disposable bottles per year. State any assumption that most affects the conclusion.
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The estimate is 620(0.45) = 279 students refilling one bottle per day. Over 180 days that is 279(180) = 50,220 bottles per year, far above 8,000. The threshold participation rate is 8,000 / (620 * 180) = 0.0717, about 7.2%. Even if only 10% of students participate, the total is 620(0.10)(180) = 11,160 bottles, still above 8,000. The conclusion is therefore robust to a fairly large error in the participation estimate. The assumption that matters most is the daily refill rate: if each participant refills only once per week, the total would be 279(36) = 10,044 bottles, still above 8,000, but if refills are much less frequent the claim could fail. The club should report that the 8,000-bottle goal is easily met under its stated assumptions, while naming the participation and refill-frequency assumptions.
3. The principal wants to report the flow rate as 'about 0.4 gallons per minute.' Is that an appropriate summary? Explain using precision.
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'About 0.4' is acceptable for a public summary but is too coarse for checking the vendor's claim. The measured value is about 0.41 gal/min, which has two meaningful figures; writing 0.4 suggests only one significant figure and could reasonably mean anywhere from about 0.35 to 0.45. That range includes the vendor's 0.40, so the summary hides the fact that the measured rate is slightly higher. For the grant application, the principal should report 0.41 gal/min, or about 0.41 +/- 0.01 gal/min, and state that this is based on a single timed trial. For a quick public statement, 'about 0.4 gallons per minute' is not false, but it should not be used to decide whether the vendor's specification was met.