40–43. Population growth

A culture of bacteri...

Question

40–43. Population growth

A culture of bacteria in a Petri dish has an initial population of 1500 cells and grows at a rate (in cells/day) of N′(t) = 100e^−0.25t. Assume t is measured in days.

a. What is the population after 20 days? After 40 days?

Accepted Answer

Welcome back, everyone. A yeast culture begins with 900 cells and grows at a rate of Y T equals 60 E raises the power of -3 divided by 20 multiplied by T cells per hour, where T is in hours. What is the population after 12 hours? Round to the nearest integer. A 1,126 cells. B 1,198 cells. C. 1,234 cells and D 1 1,298 cells. So for this problem, we first will want to identify the population function Y of T. Since we're given the growth rate, we're going to take the inside derivative of the derivative, right, because it basically returns the original function itself. So what we want to do in this problem is integrate 60E, raise the power of -3 divided by 20 multiplied by TDT. We can factor out the constant which is 60 and simply integrate E raise the power of -3 divided by 20 TDT. So now let's integrate. We got 60, which is our constant multiplied by. Let's recall that the integral of E raises the power of AXDX is equal to E raise the power of A X divided by a plus a constant of integration C. So in this case, we get E raise to the power of -3 divided by 20C. Divided by 3 divided by 20 because that's our A. And by adding a constant of integration C. Now let's simplify. This is -60 multiplied by 20 divided by 3. E raise the power of -3 divided by 20 T plus C. And now 60 divided by 3 is 20 and 20 multiplied by 20 is 400, so we get -400. E raised the power of -3 divided by 20 multiplied by T plus C. We can use the initial condition. We know that our culture begins with 900 cells, meaning Y of 0 equals 900. So when T is 0, Y is 900. We can now set this equation up and show that 900 equals -400. He raised the power of negative 3 divided by 20 multiplied by 0 plus c. We get E raise the power of 0, which is 1, so 900 equals -400 plus C. And therefore see Is equal to 900 plus 400, which is 1300. So we have our population function Y of T equals. -400. E raises the power of -3 divided by 20 T. Plus 1300. We want to find the population after 12 hours, right? So we're going to replace T with 12. We went to about with Y of 12, which is. -400. He raised the power of negative 3 divided by 20 multiplied by 12. Plus 1300. This is equal to -400. Ease the power of -36 divided by 20. We can simplify this expression as well. We can divide both sides by 4, so we get 4. I'm sorry, we got 9 divided by 5, so E is raised to -9 divided by 5. And we're adding 1300. Using a calculator, we can approximate the results that would be 1000. 234 cells. So the answer to this problem is C. Thank you for watching.

40–43. Population growthA culture of bacteria in a Petri dish has an initial population of 1500 cells and grows at a rate (in cells/day) of N′(t) = 100e^−0.25t. Assume t is measured in days.a. What is the population after 20 days? After 40 days?

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40–43. Population growth

A culture of bacteria in a Petri dish has an initial population of 1500 cells and grows at a rate (in cells/day) of N′(t) = 100e^−0.25t. Assume t is measured in days.

a. What is the population after 20 days? After 40 days?