Question

108. Draining a tank Water is drained from a 3000-gal tank at a rate that starts at 100 gal/hr and decreases continuously by 5%/hr. If the drain is left open indefinitely, how much water drains from the tank? Can a full tank be emptied at this rate?

Accepted Answer

Identify the initial rate of draining water from the tank, which is given as 100 gallons per hour. This is the starting value of the rate function, denoted as $$R(0) = 100$$ gal/hr. Recognize that the rate decreases continuously by 5% per hour. This means the rate function $$R(t)$$ can be modeled as an exponential decay function: $$R(t) = 100 	imes (1 - 0.05)^t = 100 	imes 0.95^t$$, where $$t is $$time in hours. To find the total amount of water drained over an indefinite period, set up the integral of the rate function from $$t=0 to$$ $$t=\infty$$: $$\displaystyle \$$int_0$$^{\infty} R(t) \, dt = \$$int_0$$^{\infty} 100 	imes 0.95^t \, dt. $$Rewrite the integral using the exponential function with natural logarithm for easier integration: $$0.95^t = e$$^{t \ln(0.95)}$$, so the integral becomes $$\$$int_0$$^{\infty} 100 $$e^{t \ln(0.95)}$$ \, dt$$. Evaluate the improper integral by integrating 100$$ $$e^{t \ln(0.95)}$$ with respect to $$t$$ from 0 to infinity, and interpret the result to determine if the total drained water exceeds the tank's capacity of 3000 gallons.

108. Draining a tank Water is drained from a 3000-gal tank at a rate that starts at 100 gal/hr and decreases continuously by 5%/hr. If the drain is left open indefinitely, how much water drains from the tank? Can a full tank be emptied at this rate?

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Key Concepts

Exponential Decay

Integration of Rate Functions

Limit of an Improper Integral