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Difficulty: MediumLinear Equations and Graphing

A municipal water reservoir is being drained at a constant rate. After 44 hours of draining, the reservoir contains 18,00018,000 gallons of water. After 77 hours of draining, it contains 13,50013,500 gallons of water. How many hours after the draining process begins will the reservoir be completely empty?

Answer: 16 hours

Answer

The reservoir will be completely empty 1616 hours after draining begins.
The correct answer is 1616 hours. The constant draining rate is calculated as the change in volume divided by the change in time: 13,500−18,0007−4=−1,500\frac{13,500 - 18,000}{7 - 4} = -1,500 gallons per hour. Since there are 18,00018,000 gallons left at the 44-hour mark, it will take an additional 18,0001,500=12\frac{18,000}{1,500} = 12 hours to completely empty the reservoir. The total time from the start is 4+12=164 + 12 = 16 hours.

Step-by-Step Solution

1
Calculate the draining rate (the slope of the linear function).
The rate is −1,500-1,500 gallons per hour.
The rate of change is the change in volume divided by the change in time: 13,500−18,0007−4=−4,5003=−1,500\frac{13,500 - 18,000}{7 - 4} = \frac{-4,500}{3} = -1,500 gallons per hour.
2
Determine the remaining time needed to empty the reservoir after the 77-hour mark.
It will take an additional 99 hours.
At 77 hours, the reservoir contains 13,50013,500 gallons. Draining at a rate of 1,5001,500 gallons per hour, the remaining time is 13,5001,500=9\frac{13,500}{1,500} = 9 hours.
3
Calculate the total elapsed time since the draining process began.
1616 hours
Adding the initial 77 hours of draining to the additional 99 hours needed gives 7+9=167 + 9 = 16 hours.

Key Concept

Linear word problems and finding intercepts

Alternative Method

We can model the volume of water WW as a linear function of time tt using the slope-intercept form W(t)=mt+bW(t) = mt + b. Substituting the rate m=−1,500m = -1,500 and the point (4,18,000)(4, 18,000) gives 18,000=−1,500(4)+b18,000 = -1,500(4) + b, which yields the yy-intercept (initial volume) b=24,000b = 24,000 gallons. The linear model is W(t)=−1,500t+24,000W(t) = -1,500t + 24,000. Setting W(t)=0W(t) = 0 to find when the reservoir is empty gives 0=−1,500t+24,0000 = -1,500t + 24,000, which simplifies to t=24,0001,500=16t = \frac{24,000}{1,500} = 16 hours.
Estimated Time:1m 30s
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Topics

Linear Equations and Graphing

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