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

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.

b. Find the function that gives the amount of water in the tank at any time t≥0.

Accepted Answer

Welcome back, everyone. A 2500 L reservoir is empty at time T equals 0. Water enters at a rate R of T equals 5 square root of T liters per minute. What is the function for the volume of water in the reservoir at any time T greater than or equal to 0? So for this problem, we want to find the function R of T, which represents the volume of water in the reservoir at any time T greater than or equal to 0. And we have our rate function R T because it is a derivative, we can basically take antiderivative of a derivative to return the original function R of T. So we're going to integrate R of TDT. This is integral of 5 square root of TDT. And now let's go ahead and integrate. We can factor out 5, and integrate T raises the power of 1/2 the T using the power rule. We get 5 multiplied by. According to the power rule, the integral of T raises the power of 1/2 is T raises the power of 1/2 + 1, which is 3 halves, and then we're dividing by 3 halves. We're going to add a constant of integration C because this is an indefinite integral and now let's simplify. 5 divided by 3 halves is 5 multiplied by 2 divided by 3 witches. 103 and T raises the power of three halves can be written as T square root of T. Let's add a constant of integration C and now let's identify it. We know that initially our reservoir is empty so we can show that R of 0 equals 0, meaning when time is 0. R is 0. Using this initial condition, we can set. R equals 0, and our T becomes 0 as well. So we get 0 equals 10/3 multiplied by 0, multiplied by a square root of 0 plus C. And from here we can solve for sea. And C is equal to 0. Meaning our function R of T is equal to 10/3 T square root of T. That's 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.b. Find the function that gives the amount of water in the tank at any time t≥0.

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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.

b. Find the function that gives the amount of water in the tank at any time t≥0.