Filling a tank A 2000-liter cistern is empty when water begin...

Question

Filling a tank A 2000-liter cistern is empty when water begins flowing into it (at t=0 at a rate (in L/min) given by Q′(t) = 3√t, where t is measured in minutes.

c. When will the tank be full?

Accepted Answer

Welcome back, everyone. An empty 4000 L bat is being filled with a liquid at a rate of Q T equals 6 square root of T liters per minute, with T in minutes. When will the bat be full? For this problem, we first of all want to identify the total volume function Q of T. And because we're given our wave function Q T. And it is the first derivative. We can basically take the antiderivative of the first derivative to return the original function Q of T itself, right? So we're going to integrate Q of TDT. Let's integrate 6 square root of TDT. We can factor out the constant which is 6 and integrate T raises the power of 15 DT. We can apply the power rule, we get 6, multiplied by T raises the power of 3 halves divided by 3 halves. So we take our original exponent at 1 to get 3 halves. This is our new exponent for C, and then we divide by that new exponent. And then we add a constant of integration C because this is an indefinite integral. Now 6 divided by 3 halves is 6 multiplied by 2 divided by 3, which is 4, so we got 4. T raises the power of 3 halves plus C. Or simply 4 T square root of T + C. Now we know that initially. Our bat is empty, so we can show that Q of 0 equals 0, meaning when time is 0 or initially. The amount of liquid is 0. Using this initial condition, we set Q to 0 and C to 0, so we get 0 equals 4 multiplied by 0 square root of 0 + C, and this allows us to solve for C. C is equal to 0. So that our function Q of T becomes 40 square root of T. So now we went to soul for a time when the amount of liquid becomes 4000 L. We set Q at T1, where T1 is our unknown is equal to 4000. And that we can show that 4000. Is equal to 41 square root of T1. What we're going to do is simply divide both sides by 4. So we get 1000 equals. We can use our original exponent instead of T square root of T, we're going to write T raises the power of three halves. So we get T1 raises the power of three halves. And from here we can solve for T1. We can raise both sides to. An exponent of 2/3, meaning the reciprocal of three halves, so we get T1 equals 1000, raises the power of 2/3. This is equal to cubic root. Of 1000 square, right? And essentially simplifying this expression, we can take cubic root of 1000, which is 10, and then we can square 10, right, which gives us 100. So we get 100 minutes because our units for time are minutes. Well, then we have our final answer and thank you for watching.

Filling a tank A 2000-liter cistern is empty when water begins flowing into it (at t=0 at a rate (in L/min) given by Q′(t) = 3√t, where t is measured in minutes.c. When will the tank be full?

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Filling a tank A 2000-liter cistern is empty when water begins flowing into it (at t=0 at a rate (in L/min) given by Q′(t) = 3√t, where t is measured in minutes.

c. When will the tank be full?