 Welcome back to our lecture series, Math 1050, College Algebra for students at Southern Utah University. As usual, I'm your professor today, Dr. Andrew Missildine. Now, we've been learning a lot about quadratic functions in this chapter, and quadratic functions have many, many applications in both the physical and social sciences, some of which we'll explore in this lecture 20. For example, in physics, the path of a projectile motion is parabolic, so we can use quadratic functions to model projectile motion. In economics, for example, if there's a linear relationship between price and consumption, this is sometimes called the demand function, then the revenue function will be quadratic, for which if cost is likewise linear or quadratic, then the profit function will likewise be quadratic, so this is not so unusual to use quadratic functions to help us with many problems. In this first example, we're going to actually consider the area of a garden. So imagine a gardener has 140 feet of fencing to fence in a rectangular garden, and it's always a good idea if possible to draw a picture of your situation here, whenever possible. So if you get some type of rectangular garden like the following, we wanna find a function to model the area of the garden since it's a rectangle. Now, we know that the area of a rectangle is gonna be area's length times width, so if we said something like, oh, this is the width and this is the length of the garden, we could figure that out. So how are we gonna work with this thing here? We don't know the width or the length, but we do know that our gardener has some restriction. She only has 140 feet of fencing to put around this. It's like the chicken wire she bought at Home Depot or whatever, right? And so this right here is what we call in story problems, a constraint. We don't just have infinite resources, we can't just type in cheat codes like glittering treasures or show me the money or things like that. We have limited resources and our gardener only has 140 feet of fencing. This is a constraint on the perimeter of the garden, which the perimeter can only be 140 feet. Now, as the shape of the garden is a rectangle, we know that the perimeter is gonna be the length of all four sides, so you end up with width plus length plus width plus length, which would be two width and plus two length. Notice with this, with this perimeter form, and we can actually solve from one of the variables, kind of treating this like we did with systems of linear equations, we could solve for, say, length. Notice that two times length would equal 140 minus 2w and divide both sides by two. We see that the length is gonna be given as 70 minus w. The reason this is useful is now if you come to your equation area as length times width, we could then substitute in the length function we had above and we could see that length is 70 minus the width and width. So when you distribute the width here, you're gonna get 70 minus, 70w, excuse me, minus just w squared. And so summarizing that, the area of this garden will be 70 times the width minus the width squared. And so we see there's a quadratic relationship between the width of the garden and the area of the garden. And so as this is our area function, it does make sense to kind of talk about the domain of this expression right here. Now, unlike just the algebraic functions we've done in general, when you have a story problem, you have to use the context to help us determine what is acceptable with the lengths. While w as a real number could be any number, right? There are some that don't make any sense. Like could we have a negative length to a garden? No. So this does tell us that the width of our garden does have to be greater than or equal to zero. And that's gonna tell us that for our domain, the domain here we have as a lower bound w cannot be less than zero. It could equal zero. I mean, that'd be kind of silly, but it is possible that you just make a garden, which is just a straight line up and down that's not gonna be a lot of area, but that is feasible, right? So zero is our lower bound. What about upper bound? Well, it turns out that the upper bound is gonna come from the other extreme, right? That the length should be greater than or equal to zero as well. And so we consider this equation that we saw up here. What would the length being zero represent? Well, if the length was zero, then we would just move w to the other side. We see that w would equal 70 when length is at zero. And so that's gonna give us an upper bound right here that the width cannot be bigger than 70 because they now would require the length to be negative. So we now have this function, a quadratic function that gives us the area of this garden. Now for our gardener, we wonder, could she make a garden whose area is 1,250 square feet? Now, what does that mean exactly? So if we're trying to figure out, is this an acceptable area, we're trying to solve the equation, area equals 1,250 square feet, which as we saw previously, we have an equation 70w minus w squared. Now in terms of solving, we don't necessarily need the whole solution. We mean though we're just like, is there a solution? Is there a solution? Is there a solution to this equation? So like we saw before, we're trying to look for the nature of the solution set. Do we have one solution, two solutions or no solutions? Because okay, our solutions would have to be real numbers in this situation. And so we're trying to determine what's the nature of the solution set for which the discriminant is actually a very helpful tool in this situation. The discriminant remember is b squared minus 4ac. It's the radicand inside of the quadratic formula. Using the numbers from the quadratic formula above, we get that b is 70. So you're gonna get 70 squared minus four times a, which is negative one. And then you're gonna get c, which is, well, I guess we got a little ahead of ourselves, right? Cause when you look at this quadratic equation to use the discriminant, we have to have everything set equal to zero. So we actually need to move the negative 1250 to the other side. So really the equation we're trying to solve is negative w squared plus 70w minus 1250 equals zero. So if we were cautious, if we were a little bit careful here, if we were careless here, we might have thought that c was zero. But in fact, we see that zero is going to be negative, sorry, c is gonna be negative 1250, like so. And so then computing this thing, negative 70, well, 70 squared is going to give us a positive 4,900. You get a triple negative here. So you're gonna get minus four times one is just four. And then four times 1250 is gonna be 5,000. So we see the discriminants can turn out to be negative 100. Negative discriminants here would mean that while there are non-real solutions that are complex, there's no real solution. And our gardener needs to make a garden which has real boundaries, a real width, a real length. And so what this tells us here is that for our gardener, that there is no solution to the equation. So for her, that tells us that she cannot fence a garden with an area of 1250 square feet. Now it's not because she's incompetent at putting the chicken wire down. It really just comes down to that with the constraint of 150 feet of fencing, 1250 is too big. You cannot make a garden that big. So she has a choice to make. She either has to run back to her local farm supply store or garden supply store or home supply store, right? And get some more fencing or she has to accept that 1250 is too big. In which case she kind of turns to the question, oh, well, okay, how big of a garden could she make? What is an acceptable size? What's the biggest size? And it leads very naturally to that question. What is the largest area that she can fence? What is the biggest value? Now, if we look at our function, A equals 70w minus w squared. I want you to notice that the leading coefficient here is negative one. It's a negative. So if we were to graph this thing with versus area, our graph would look like, it would look like a downward concaving parabola for which you have w equals zero as the left x intercept and the right left, the right x intercept would be zero. Sorry, 70 would be the width. Both of these would be situations where the area is zero but there should be a maximum value right here which is maximum value would coincide with the vertex of the parabola. So the maximum area can be obtained by looking at the vertex. So as we've seen before, the vertex, the x coordinate can be found as h equals negative B over 2a which using the values we have right here, we get a negative 70 over two times negative one that gives us 35 feet. Now, this is not the maximum area. This is the dimension of the rectangle that will make it the maximum. So this is saying that the width should be 35 feet. So if the width is 35 feet, how big should the length be? Like we saw before, the length is equal to 70 minus w which in this case would be 70 minus 35 which is likewise 35 feet. So we see that the maximum garden would be obtained by 35 feet by 35 feet. So it turns out for our gardener, she would be advantage to create a square garden because that would maximize the area. But how big would the maximum area actually be? The maximum area is gonna be obtained when x or when w here equals 35. So we get 70 times 35 minus 35 squared. So you can compute that. But be aware as the area is just length times width that we already figured out what the length was. This is gonna be 35 times 35, 35 squared that is. It's a square, right? And so this would give us a maximum area of 1,225 square feet. So it's a little bit shy of the 1250 square feet she was looking at earlier. So again, she could get a little bit more fencing if she wants to get 1250, but if she's just gonna stick with the fence and she has right now, she should make a square garden and that maximum area would be 1,225 square feet.