A tank has 30 gallons of water in it at time t = 0 hours....

A tank has 30 gallons of water in it at time $t = 0$ hours. Water begins to be pumped into the tank at time $t = 0$. A different pipe is draining water from the tank starting at $t = 0$. Water is being removed from the tank at a constant rate of 10 gallons per hour. Water is being pumped into a tank at a rate modeled by the function $M(t) = -9(0.82)^t + 12$, where $M(t)$ is measured in gallons per hour. What is the minimum amount of water in the tank from $t = 0$ to $t = 11$? You may use a calculator and round to the nearest thousandth. Use proper units and show how you got your answer.

Transcript

00:01 All right, for this question, we are given the equation for a rate of water being poured into a tank.

00:07 We're given a rate in as well as a rate out.

00:11 We're also told the initial amount of water, which is 30 gallons.

00:17 And our goal is to find the minimum amount of water from 0 to 11 seconds.

00:22 So to do this, we're going to find a critical point.

00:25 And to find a critical point, you set the rate equal to zero.

00:32 And the rate that we're looking at in particular is going to be the rate in minus the rate out, which means we're going to have the negative 9, the 0 .82 to the t.

00:45 We'll have plus 12 minus 10, so that will be plus 2.

00:52 And we have the set equal to 0.

00:54 And now we have an equation where we can solve for t.

01:02 So subtract the 2, divide out the negative 9, take a natural log from each side, this will allow the t to come out of the exponent and come out front as a multiple.

01:28 And then to finally solve for t, you can divide out the ln of 0 .82.

01:42 And there we go.

01:43 This comes out to roughly 7 .579 seconds, which is going to be a critical point since it falls in our interval from 0 to 11.

01:59 And what we also would like to do now is to get an equation for the amount of water in the tank at any given time.

02:10 To do this, you'll start with the initial amount of water, and then the net change is going to be represented by an integral from zero to whatever value of t we go up to.

02:25 And we'll have our rate function inside the integral.

02:38 And perhaps to avoid confusion, you should usually have a dummy very variable.

02:41 Variable inside the integral...

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