Quadratic word problem: mosquitoes

- [Voiceover] The number of mosquitos in millions, M, so M is the number of mosquitos in millions. In Brooklyn, New York it depends on the June rainfall in centimeters, R. So R is the amount of rainfall measured in centimeters. And can be modeled by the function M, so the number of mosquitos in millions, is equal to negative R times R minus four, where R, once again, is the rainfall in centimeters, I guess in June. Alright, fair enough. Now let's see if we can answer these questions. And like always, I encourage you to get inspired and try to answer these questions on your own. (laughing) Alright, first one. How many centimeters of rainfall will there be... After how many centimeters of rainfall will there be a population of zero million mosquitos? Or will there be just no mosquitos? Zero mosquitos. Well let's just rewrite this function. So our population of mosquitos, it actually just helps my brain to rewrite it, to kind of process what's going on. My population of mosquitos in millions is going to be negative R times R minus four, where R is the number of, the number of centimeters of rainfall. So in order to have a zero million population, so that means M is going to have to be equal to zero. So we have to figure out at what Rs, at what Rs is M equal to zero? So we just have to solve this right over here. Well in order for this expression right over here to be equal to zero, notice it's the product of two expressions. We are multiplying, you could view it as we're multiplying negative R times R minus four. Let me do R minus four in a different color. Times R minus four. So if you're multiplying two expressions and you get zero, one or both of them need to be equal to zero. So let's see how that happens. So either negative R is going to be equal to zero, negative R is equal to zero, or or you're going to have R minus four is equal to zero. Or R minus four is equal to zero. So how does negative R equal zero? We divide both sides by negative one or multiply both sides by negative one and you get r would need to be equal to zero. How does R minus four equal zero? Well you add four to both sides, you see that R would have to be equal to four. So it either is no rainfall, zero centimeters of rainfall, or four centimeters of rainfall, you get a population of zero mosquitos. Now why does that, let's just think about whether that makes sense. Of whether this is a good model for for the number of mosquitos. Well we always imagine mosquitos need some standing water, so if you have no rainfall, there's not going to be any place for them to breed. There's not going to be any standing water, maybe. And the way this model seems, it's like well if you have enough rainfall, so some rain is gonna help the mosquitos breed, but then if you have enough rainfall, they're just gonna get I guess washed out someplace. i guess raindrops are gonna hit them (laughing) as they fly, I don't know what the, what's actually happening at the mosquito scale, but it's implying that if there's a lot of rainfall it would actually be hard. I guess you don't have standing water any more. The water is moving around and is overflowing and it's going into the gutter and whatever else. It would make it vey hard for the mosquitos to breed. What is the maximum possible number of mosquitos? So the very idea that they're asking us a maximum implies that this quadratic over here, and it is a quadratic and we'll see that more clearly in a few seconds, is shaped like this in order to have a maximum point. Because if it was shaped like this, it would have a minimum point, it wouldn't have any one maximum. And because we have this negative out here, that makes us pretty confident it is going to be a downward opening parabola. But to understand what this maximum point is, let's write it in vertex form. Just as a reminder what vertex form would be, it would be writing it in the form of M is equal to A times R minus B squared, plus C. This right over here would be vertex form and it's very useful because you get a sense of when you hit a minimum or maximum point. If thing right over here is negative, then the maximum value, because this thing can only be non-positive, the maximum value would be this, and then we could figure out when does this equal zero? And that's when you hit that maximum value. And if any of this seems foreign, I encourage you to watch the videos on vertex form. And I'm about to complete the square on this character in order to get here, so I encourage you to also watch the videos on completing the square. So let's do that. So let's start with this expression and I'm gonna start by, actually, let's start by, I'm gonna just distribute the R. So if you have M is equal to, I'm gonna keep the negative out front, but I'm gonna distribute the R. R times R is R squared. R times negative four is minus four R. And I want to complete the square. And once again, I encourage you to watch those videos if what I'm about to do seems like voodoo, but to complete the square I want to add and subtract the same thing. I don't want to actually change the value of the expression. And so I'll look at this coefficient, this first degree coefficient here, it's negative four. Half of that is negative two. If I square it, I want to add four. I want to, I want to add four. I want to add, let me make it a different color. I want to add four. Now once again, I can't just go around adding numbers to expressions and assuming the equality still holds. I have to add and subtract the same thing. While you might be tempted to write a minus four out here, because hey, I added four and then I subtract four. But remember, when I add this four in this parentheses, i have this negative out front. If I were to distribute this negative, I essentially, this is a minus four. So if I wanted to add them out to be zero, distribute this negative, this would be a negative four. If I wanted to cancel out this, this needs to be a positive four. So once again, when you look superficially, it looks like I just added four twice, but I didn't. this right over here since it has this negative sign, is a negative four. Negative four plus four is zero. I have not changed the value of this expression. But anyway, now that I've done that, this piece right over here, the whole reason why I added four is to make it a perfect square, this is now the same thing as R minus two, R minus two squared. I have the negative out front. We have M equals negative R minus two squared plus four. Plus four. And just like that, I've written it in vertex form. So what's the maximum number of mosquitos? Well let's just look at the structure of this expression a little bit. This piece, this piece right over here, that thing, actually, let me not do it in the same color as the four, let me do it in this orangish color. This piece right over here is always going to be less than or equal to zero. So that's always going to be less than or equal to zero. It's always going to be non-positive. And so this whole expression hits a maximum value when this thing is zero, because at any other point, it's gong to just take away from this right over here. So you hit a maximum value when this is equal to zero. When negative R minus two squared is equal to zero. Well that's going to happen when, well you can see when R minus two is equal to zero. Or when R is equal to two. And when R is equal to two, what is M going to be equal to? Well when R is equal to two, this is zero and M is equal to four. R equals two and M is equal to four. So what's the maximum possible number of mosquitos? Well M equals four, it's gonna be four million mosquitos. Remember, M is measuring in millions. Four million mosquitos. And then they ask us after how many centimeters of rainfall will the maximum number of mosquitos occur? Well, we figured it out. After two. After, after two centimeters. Two centimeters of rainfall. So if we go back to our, if we go back to our model, at zero rainfall you have no mosquitos, then as it rains more and more, you have more and more mosquitos. I guess more and more standing water. You have a maximum number of mosquitos at two centimeters of rainfall. At two centimeters we have four million mosquitos, but after that I guess the rain starts washing away the mosquito eggs or something and you start having fewer and fewer mosquitos all the way until you get to four centimeters of rainfall. And I guess after four centimeters of rainfall, in theory you have negative mosquitos, I guess. I don't know what that would mean, and that might just be a situation where the model just breaks down. You obviously, I don't know what negative million mosquitos would actually mean. The relevant, the relevant area where this model should hold is between zero centimeters of rainfall and four centimeters, because after that the model starts describing negative values for M. So anyway, hopefully you found that interesting.