Suppose a population of mice on an island grows...

Suppose a population of mice on an island grows exponentially according to the mathematical model $y = y_0e^{kt}$ where $t$ is measured in weeks. At time $t = 0$, there are 300 mice. After 2 weeks there are 600 mice. After 6 weeks there are 2400 mice. If using the above information, which of the following statements are correct? Select all that apply. $y_0 = 600$ $6 = 2400e^{kt}$ $y_0$ can not be determined with the information given. After 3 weeks there are 1050 mice. $300 = 600e^{2k}$ $600 = 300e^{2k}$ $y_0 = 300$ $2400 = 300e^{6k}$ $2400 = 6e^{600k}$

Transcript

00:01 Hi there, so for this problem, we are given the following differential equation that the population of rabbits, the derivative of that with respect to time, is equal to a concept of proportionality times the population at any given time to the square.

00:22 Now, we are given the information that the initial population of rabbits is five rabbits, and it is increasing at the rate of two rapids per month.

00:34 So we're given that the real change of the population of rabbits with respect to time when the population p is equal to 25, then this is equal to two.

00:51 Now, once we note this, we need to determine which, if any of the following statements is correct.

01:00 So what we need to do first, and with this information in here, we can first determine the value of the concept of proportionality k.

01:10 Because by using this condition, we will have that this is 2 is equal to k.

01:17 This when the population is 25 to the square.

01:22 So that will give us that the constant of proportionality is 2 divided by 25 to the square.

01:30 So that will be 25 to the square is equal to so we will have that that is 2 divided by 625.

01:41 So with that, we will have that now.

01:46 The differential equation is equal to 2 divided by 625, this times the population square.

02:01 Then we can just separate the variable.

02:04 So we will have the differential in the population divided by the population to the square is equal to 2 divided by 625 times the differential in time.

02:18 Now we can integrate both sides of this expression.

02:22 Now for the left side we obtain minus 1 divided by the population.

02:35 Then this is equal to 2 divided by 625, this times the time, this plus a constant of integration, because this is an indefinite integral.

02:54 Now to find the value of c, we can use the initial condition that the population at the time equals to 0 is 5, so we will have minus 1 divided by the population.

03:07 Is equal to c, well, this is when the population is equal to 5, the initial population.

03:21 So from this, we obtain that c, it hits this value, so we will have minus 1 divided by the population is equal to 2 divided by 625 times the time this, minus 1 divided by 5.

03:36 So we can simplify this further, so we will have that this is minus 1 divided by the population is equal to, so we hope that this is 2 times 5, so that will be 10 times the time, this minus 625.

03:58 This divided by 625 times 5, so that will be 3 ,000 ,000, 125.

04:11 We can divide all of these numbers by 5...

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