108. Draining a tank Water is drained from a 3000-gal tank at ...

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

Welcome back, everyone. A 3000 g radioactive substance decays at a rate of 150 g per hour, with the rate decreasing by 6% per hour. If the decay continues indefinitely, what is the total mass that will decay? Can all 3000 g decay in this process? Round the answer to the nearest whole number. For this problem we're going to define our function. What we're going to do is simply express our rate function as a function of time. We have the initial rate R 0 equals 150 g per hour, right? Now rate decreases by 6% per hour. Therefore, at time T, we can write. R of T equals 150 multiplied by 0.94 to the power of T. So we're going to get that exponential function. We essentially subtract 6% from 100% to get 94% of the previous rate, right? And this essentially gives us the decimal expression of 0.94. We are raising it to the power of T and multiplying by the initial rate of 150. So, for example, if T is equal to 0, we get the initial rate. If T is equal to 1. We get 94% of the original rate. If T is equal to 2, we're going to square 0.94 and multiplied by the original rate to get the rate in 2 hours and so on. So we're going to use this function in our analysis to find the total mass decayed. Let's call it M. And specifically we can use lowercase m, right? Let's remember that if we want to get the total mass, we have grams in the numerator, right? We essentially want to integrate the rate function, starting from the initial time of 0 up to infinity because it says indefinitely. We're going to have that improper integral and we're going to integrate R of TDT. Simply speaking, we're integrating 150 multiplied by 0.9 or to the power of TDT from 0 up to infinity. What we can do is factor out the constant 150. And integrate from 0 to infinity, 0.94 to the power of TDT. Let's recall that the integral from 0 to infinity of an exponential function A to the power of TD T is equal to -1 divided by LN of A when A is between 0 and 1, not inclusive, and this is the case here, right? 0.94 is between 0 and 1. So we can show that our integral is 150. Multiplied by -1 divided by LN of 0.94. What we're going to do from here is perform the calculation and round our answer to the nearest whole number. We get approximately 2420 g. Meaning All 3000 g cannot decay in this process, so the second answer to this problem would be no, since our number 2,424 g is lower than 3000. So let's write our answers 2,424 g and no. Thank you for watching.

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?

Key Concepts

Exponential Decay

Integration of Rate Functions

Limit of an Improper Integral

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