00:01
This problem focuses on an apartment manager who is trying to figure out how much to rent the apartments for.
00:09
He has 80 units available to be rented.
00:14
So our variable is going to be x.
00:16
So we're going to let x equal the number of $20 increases to the rent.
00:27
Because let's take a look at what we have.
00:29
At $400 a month, every unit is rented.
00:33
How many apartments are rented if i make x increases? well, for every increase, i lose one apartment that's being rented.
00:43
So one increase means i have 79 rented.
00:46
Two increases is 78.
00:48
Three would be 77 and so on.
00:50
So this is the number of apartments that will be rented.
01:01
Now, how much rent will he get per apartment? well, he's starting with 400.
01:08
And we're putting in so many increases.
01:13
Each increase gives him an additional $20.
01:17
So one additional increase will be $420.
01:21
Two increases is $440.
01:23
Three increases is $460 and so on.
01:26
So this says how much the rent is per unit.
01:35
And we can use these two pieces of information to find the revenue function that he will get for these units.
01:41
Because his revenue is going to be the number of units he rents out, which is 80 minus x, times the rent per unit, which is 400 plus 20x.
01:58
If i multiply this out, that gives me 32 ,000 plus all the outer and the inner is 1 ,200x minus 20x squared.
02:13
And just for simplicity sake later, i'm going to rewrite this in descending order.
02:27
So there is our revenue function.
02:30
We can use this to find all sorts of scenarios.
02:33
How much money will i make if i rent out this many units? if i have this much as my rent, what does this give me? what's my maximum rent? lots of questions that we can answer with this.
02:45
Two, what we're going to look at in particular.
02:48
First, for what number of increases? will the revenue be 37 ,500? so i want the revenue to be 37 ,500.
02:59
And i want to solve for x, the number of increases that will be required to meet this number.
03:08
Now, to do this, we're going to set everything equal to zero.
03:12
So i'm going to pull everything over.
03:14
Actually, i'd like to have everything positive.
03:16
So i'm going to pull everything over to the left -hand side of this equation.
03:21
I like to have a positive x squared term.
03:23
So, 20x squared minus 1 ,200x plus 5 ,500 equals 0.
03:35
We can make these numbers a little smaller.
03:38
Everything here is divisible by 20.
03:40
So that gives me x squared minus 60x plus 275 equals 0.
03:49
Fortunately, this is a factorable trinomial, and it factors into x minus 5 and x minus minus 55...