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Problem 4 Hard Difficulty

A bacteria culture grows with constant relative growth rate. The bacteria count was 400 after 2 hours and 25,600 after 6 hours.
(a) What is the relative growth rate? Express your answer as a percentage.
(b) What was the initial size of the culture?
(c) Find an expression for the number of bacteria after $ t $ hours.
(d) Find the number of cells after 4.5 hours.
(e) Find the rate of growth after 4.5 hours.
(f) When will the population reach 50,000?

Answer

a) $k=\frac{\ln (8)}{2}$ . Relative growth rate is 104$\%$
b) The initial size of the culture was 50
c) Number of bacteria cells after $t$ hours will be governed by $A(t)=50 \cdot 8^{0.5 t}$
d) The number of bacteria cells after 4.5 hours are 5381
e) The rate of growth of bacteria cells after 4.5 hours is 5595 cells per hour
f) The population of bacteria will reach $50,000$ in 6.65 hours

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Video Transcript

in this problem were given some initial information about the bacteria count in terms of hours and the number of bacteria. After two hours, there were 400 bacteria and after six hours there were 25,600 bacteria. And we're going to use the population growth model. P equals P, not each of the K T and for party. The first thing we want to do is find the relative growth rate, and that's what we call K. So what we can do is substitute some of the given numbers into this function and find K. So let's let the final amount pft B 25,600 and the initial Mt. P not is 400. And the time that has elapsed would be four hours from time to time. Six. So we're going to let t equal for So we have each of the four times K and we're going to solve this equation for K. So we're going to start by dividing both sides by 400 we get 64 equals each of the four K and then we're going to take the natural log on both sides. So natural log of 64 equals four K, and then we're going to divide both sides by four. So K is the natural log of 64/4. Now, that is accurate, but it can be reduced if you think about 64 as eight squared. So with the natural log of eight squared over four and then you think about you or longer than properties and how you can bring your exponents out to the front. So this would be equivalent to two times the natural log of 8/4. And if you reduce the two in the four, you get a K value of the natural log of eight over to okay. It says in the problem that we're to express this as a percent. So first, if you put it into a calculator, you get approximately 1.4 so as a percent, that would be 104%. Okay, let's move on to part B. What we want to find in part B is the initial size of the culture. So we're finding P. Not so. What we can do is go back to our equation. P of t equals p, not each of the K T. Now we know the value of K. We can take one of our ordered pairs. I'm going to use the point to calm a 400 substituted in and solve for P. Not so the final value is 400. We don't know the initial value. P not. We know the value of K is natural, like eight over to, and we know the value of tea at this moment is to so we'll solve that for peanut notice. We can simplify the exponents. So now we have 400 equals p not times each of the natural log of eight and notice that each of the natural log of eight can be simplified. That's just eight. So at this point, we have 400 equals P not times eight. We can divide both sides by eight, and we get 50. So the initial population is 50. Okay, let's go ahead and move on to part C. And in part C, we put all this information together and write the expression for the number of bacteria after, um, t hours. That basically means just put the model together. So we take our model p of t equals p not each of the Katie. We substitute 50. And for Peanut. We just found that number and we substitute the natural log of 8/2 and four K. We found that number earlier in part A. And here we have our model. Now, we could actually simplify this a little bit, because ah, we can take E to the natural log of 8/2 times t and simplify it in the following way. We could rewrite that as he to the natural log of eight times 1/2 a T or 0.5 t and then thinking of it as a power to a power. You multiply the powers we can think of it as each of the natural log of A to the power 0.5 t and each of the natural log of eight is eight. So we have eight to the power 0.5 t. Okay, So putting that into our model that we were just writing, we have p of tea equals 50 times eight to the 0.5 t so we could use that model moving forward as we do. Part e r. Part D. Excuse me. So here we are doing Part D. Here's our model P A of t equals 50 times eight to the 0.5 T, and in this part we want to find the population at time. 4.5. So let's find P of 4.5. So we substitute 4.5 into this equation and we get approximately 5381.7, which I would round 2 5382 But the book says 5381. So is there a reason for that? It could be that in the book, whoever did the solution rounded some numbers early, and when you do early rounding than your final answer is not as accurate. So that's what I'm speculating, but I don't know for sure. Okay, so now we'll move on to party where were asked to find the rate of growth. Now the rate of growth would be the rate of change, and that would be the derivative. So let's find the derivative of the function we just used in party. So we're finding the derivative of an exponential function. Remember, the derivative of A to the X is a to the X Times natural log A. So this is going to be 50 times eight to the 0.5 tee times natural log eight times the derivative of the inside, which would be 0.5. Okay, putting all of that together if we want, we could simplify it or we can just move forward and substitute the 4.5 in there. We're looking for the rate of growth at time. 4.5. Put that in the calculator and you get approximately 5595.5, which I would round to 5596. But the book says 5595 now again. Is that because they used around in value earlier? Or is that because they're just choosing to round down? I don't know for sure. Okay, we have one final part to do in this problem. Part F in Part F says when will the population reach 50,000? So we know the population we don't know the time, so lets you their model again. P of tea equals 50 times eight to the 0.5 t. Let's substitute 50,000 in for the population and saw for tea Divide both sides by 50 and we get 1000 equals eight to the 0.5 t take a natural log on both sides. Use the power property of lager thems to bring that 0.5 t down live natural log 1000 equals 0.5 tee times natural August 8 And then to isolate T. We're going to divide both sides by 80.5. And we're also going to divide both sides by natural log eight. And then this goes into the calculator and we get approximately 6.64 hours. The book has 6.65 All right, there we go.