40–43. Population growth

When records were fi...

Question

40–43. Population growth

When records were first kept (t=0), the population of a rural town was 250 people. During the following years, the population grew at a rate of P′(t) = 30(1+√t), where t is measured in years.

b. Find the population P(t) at any time t≥0.

Accepted Answer

Welcome back, everyone. A village had a population of 150 at T equals 0. The population growth rate is given by P T equals 25 multiplied by 1 + 2 square root of T with T in years. Find the population P of T for T greater than or equals to 0. For this problem we want to identify P of T, which is integral of P TDT, right? The anti derivative of a derivative returns the function itself. So what we want to do is simply integrate 25 multiplied by 1 + 2 square root of TDT. Let's go ahead and factor out the constant, that's 25 and Cs. We can split our integral into two parts. First of all, that's integral of 1 DT or simply DT. And then we're adding 2, we can factor out the constant integral of square root of T, which is T raises the power of 1/2 DT. Now let's go ahead and integrate. That's 25 and Cs. The integral of the T is T. We're going to add 2 multiplied by the integral of T raises the power of 1/2 is T raises the power of three halves divided by 3 halves according to the power rule. And then we're adding a constant of integration, see because this is an indefinite integral. Now we can distribute, we got 25T plus. 2 divided by 3 halves is going to be 2 multiplied by 2 divided by 3, so that's 4/3. And we get 25 multiplied by 4/3. C raises the power of three halves can be written as T square root of C, and we're adding a constant of integration C. Simplifying, we get 25 T plus 100 divided by 3. Multiplied by T square root of T + C. And that we can use the initial condition P of 0 equals 150, which essentially means that if T is 0, P is 150. We can set 150 equals. 25 multiplied by 0 plus. 100 divided by 3 multiplied by 0 multiplied by square root of 0 plus C. From here we can see that C is equal to 150. And now we have the required function, PF T equals 25T. Plus 100 divided by 3. T square root of T + 150. That's our final answer and thank you for watching.

40–43. Population growth

When records were first kept (t=0), the population of a rural town was 250 people. During the following years, the population grew at a rate of P′(t) = 30(1+√t), where t is measured in years.

b. Find the population P(t) at any time t≥0.

Key Concepts

Derivative as a Rate of Change

Antiderivative and Indefinite Integration

Initial Conditions and Constant of Integration

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