 This video will talk about scientific models. The door handle starts out with 1500 bacteria, so that's our beginning amount. And this strain of bacteria is known to double every 8 hours. Find the growth rate of this bacteria. Well, to find the growth rate, we need a formula. The quantity that we're going to end up with is equal to the quantity that we started with, so Q of 0, times E to the RT. And we are going to say that this is our Q original. So we have 1500 here, times E to the R, which we don't know, that's what they asked us to find, times 8. And all we need over here then is the idea that it doubled. So I have twice my Q0, which is 1500. So again, when I divide off the 1500, I'm going to have 2 is equal to E to the 8R. And we just have to take the natural log of both sides. And again, bring the exponent down times my natural log of my base. And if you remember, we said that this was really just like multiplying times 1. So my final answer here then should be ln2 divided by 8 is equal to my rate. And that's good enough. I don't really care exactly what it is, I just need a representation of that rate. So then when we do part B, we know that R was equal to ln2 divided by 8. So how long will it be until the door handle has 30,000 bacteria? Assume that the door handle research purposes is not touched. So it's going to grow naturally. We want to know how long that it will then get to this. So this would be our Q of T. And if you remember, our Q of 0 was 1500. So if we solve our equation, then we have 30,000 is equal to our 1500 times E to the R, which now is ln2 over 8 times T. Well dividing the 1500 off, we're going to have 20 over here, equal to E to the ln2 over 8 times T. And natural log, you're right. Natural log of 20 is equal to ln2 over 8T times lnE, which is just 1. So we have finally ln20 divided by ln2 divided by 8. And that will be equal to our T. And if we take that to our calculator to find out exactly how much time that means, we have ln20 divided by, and then I'm putting parentheses around the denominator since it's got an operation in it. So ln2 close the argument divided by 8, and then use a parentheses to close the denominator. And we find out that it's going to be about 34 and a half months, a little bit more than that. So approximately 34.6 months. So the next time a scientific problem that we like to do is talk about half-life. And I want to work through this for you, and we're going to try to figure out rate. But once we figure out a formula for rate, we won't have to do this again over and over and over again. So here we go. We have the original amount Q of 0, and this Q of T is the final amount. And it's E to the negative RT. Negative RT is for a decay. And if it were a positive RT, which is what we did with the problem before, that is a growth. So we have a negative RT happening here. And then we're talking about half-life. So that means we're going to have half of the original amount, so half of Q0. And that's equal to our Q0, E to the negative R, and then H. We're going to replace T with H to let it be the half-life, because that's something we will know. So if we simplify that, we can divide off the Q of 0. So we have 1 half is equal to E to the negative RH. And if we take the negative exponent, we could write it as 1 over E to the RH. So 1 half is equal to E to the RH. Or we could really kind of just look at this part down here then, since they're both in the denominator, we could say that 2 is equal to E to the RH. And taking the natural log of both sides so that we can get sulfur that R, natural log of 2 is equal to RH times the natural log of E. And if we want to solve, remember this is 1, so we want to solve for R, we just have to divide L into by H. So this is going to be our normal rate of decay per unit. We have this problem of sodium 24. And before we go any further, I want you to recognize that this is just a substance. That 24 is not a number, it's just a name. Sodium, which sodium is sodium 24? It's a radioactive element that has a half life, so that's going to be important, of 15 hours. It's used to help locate obstructions and blood flow. And if the procedure requires 0.75 grams and is scheduled to take place in two days or 48 hours, we want to know. What we're trying to find is how much has to be on hand now. So let's see what we have here. Half life, that means that we're going to have half of our original Q0. 15 hours, the half life mean 15 hours, that means that H is equal to 15. And we have 0.75 grams, that's going to be the amount that we want to end up with. So that's our Q of T. And then the 48 hours is going to be a T. Remember the formula is Q of T is equal to Q of 0. E to the negative R, and we will call that T. So we know that R is going to be equal to LN2 divided by H. Or in our case, LN2 divided by 15. So let's plug and show what we know. 0.75 is our Q of T up there. Q of 0, we don't know what that is. That's what we're trying to find out because that's the now. That's Q of 0 right here. And then E to the negative R. But our R now is LN2 over 15 times our T, which we know is 48. So we have everything except for Q of 0. And that's what we have to solve for. We want to get to this thing. So how do we do that? So we can divide 0.75 divided by that E. E to the negative LN2 over 15 times 48. All of that is my exponent is equal to Q of 0. That's what we want. So if I call it my calculator and bring it over 0.75 divided by And then it's second LN. That gives me E. And then in my carat, I have negative LN2. Close the parentheses divided by 15. And then I'm going to need an extra set of parentheses and times the 48. But I need a double one here. And I really needed to go up here and put one, a double one, second insert or second delete will insert one. So if I insert a parentheses, then now I have LN2, negative LN2 over 15 is one of my factors times 48. And the whole thing is my exponent. And I find out that I started out with about 6.89 or 6.9. So approximately 6.9 grams.