This problem focuses on an apartment manager who is trying to figure out how much to rent the apartments for. He has 80 units available to be rented. So we are variables gonna be X. So we're going to let X equal the number of $20 increases to the rent. Because let's take a look at what we have. At $400 a month, every unit is rented. How many apartments? Air rented If I make X increases well, for every increase, I lose one apartment that's being rented. So one increase means I have 79. Wretched to increases in 78. 3 would be 77 so on. So this is the number of apartments that will be rented. Okay, Now, how much rent will he get? Per apartment. While he's starting with 400 and we're putting in so many increases, each increase gives him an additional $20 so one additional increase will be 422 increases is 4 43 increases is 460 so on. So this says how much the rent is per unit. We can use these two pieces of information to find the revenue function that he will get for these units because his revenue is going to be the number of units he rents out, which is a T minus x times the rent per unit, which is 400 plus 20 x. If I multiply this out, that gives me three. 32,000 plus all the outer in the inner is 1200 X minus 20 X squared. And just for simplicity sake later, I'm gonna rewrite this in descending order. So there is our revenue function. We can use this to find all sorts of scenarios. How much? How much money will I make if I rent rent out this many units? If I pep this much as my rent, what does this give me? What's my maximum rent? Lots of questions that we can answer with this to go and look at in particular First, for what number of increases will the revenue B 37,500. So I want the revenue to be 37,500 and I want to solve for X number of increases that will be required to meet this number. Now to do this, we're going to set everything equal to zero. So I'm gonna pull everything over. Um, actually, I'd like to have everything positive. So I'm going to pull everything over to the left hand side of this equation. I like to have a positive X squared term. So x 20 X squared minus 1200 X plus 5500 equals zero. Okay, We can make these numbers a little smaller. Everything here is divisible by 20. So that gives me X squared minus 60 X plus 2 75 equals zero. Fortunately, this is a fact. Herbal. Try no meal and it factors into X minus five and X minus 55. So we actually have two different scenarios. If I set these equal to zero, I either get X equals five or X equals 55. So in order to make my 37,500 as my revenue target, I either need tohave five increases of $20 or 55 increases of $20. Either one of those values will give me the revenue I'm seeking. Now, what is my maximum revenue? Well, looking at this equation back here that start equation the one that starred I put a blue star next to it. If you examine that, you can see that the highest degree term is X squared, which means this is a parabola. It has a negative coefficient, which means it's a downward facing parabola. So to find my maximum revenue, I need to find my Vertex. So let's come. I'm going to scroll up, just gonna scroll up just a little bit, do that just a little bit. That gives me a little bit more space here. I'm going to take my revenue function and it just re copying it. So I have some space toe work underneath it. And in order to find the Vertex, the easiest way to do that is to complete the square. So every term that has an accident, I'm going to factor out a negative 20 because I want that X squared term toe have a coefficient of one. So I factor out a negative 20. So I get X squared minus 60 x, leave myself some space to work, and I take my constant term and put it over on the side. It could just sit there for a little bit Now to complete the square. We need to look at the coefficient of R X term. It's negative. 60. I want half of that, which is negative. 30. And when I square it, that's what I'm gonna put back up into my equation. Plus 900. No, I can't just add 900 so I need to subtract 900 as well. That means I haven't made any fundamental changes to my equation. So writing that here I get X squared minus 60 x plus 900 that's my perfect square. This piece I need to move outside of my parentheses, remembering that everything in the parentheses is being multiplied by negative 20. So when I pull it out, I am actually adding 18,000, which I'll combine with the 32,000 that's already out here. The final form of this function then is going to be negative 20 times X minus 30 square. There's my perfect square. I was going for plus 50,000. This is the Vertex. I can pull it right off of this form. The Vertex is going to be at the 0.30 50,000. So what does this number actually mean? Remember, this is my ex, So that means I need to have 30 increases of $20. 30 increases of $20 gives me an increase of $600 over my initial starting point of 400. So that means I'm going to be needing to charge $1000 per apartment, leaving me by maximum possible revenue of $50,000 total.