00:01
So we're told that we want to maximize revenue.
00:05
Revenue occurs when we multiply the price of a product times the number of products that we sell.
00:11
In this case, we're discussing an apartment complex.
00:14
And so we're told that from experience, if 110 units are rented out at a rent of 400, would be rented at $400.
00:21
And then we're told that for every $1 increase in the rent, so $401, one unit would become vacant.
00:28
So then we would be down to 109 units being rented out.
00:32
If i did that again, i could charge a rent of $402, but i would only have 108 units being rented out.
00:38
And so again, we can think of the revenue as multiplying the number of units rented out times the rent price.
00:44
And what we want to do is maximize the revenue.
00:46
So how many units do we need to rent out and what would the price of those units be? so we're going to write a revenue function.
00:53
X is going to be the number of times that we increase the rent by a dollar or decrease the rent by a dollar because it'd be the same in the opposite direction.
01:02
And so we can think of starting with 110 units.
01:06
And with every increase in x, so if i were adding x over here, so plus $1, i'm subtracting x over here.
01:14
So we can say 110 minus x.
01:16
Again, x being the number of times i increase the rent by a dollar.
01:20
That part of the function would then be multiplied by $400 because that's my rent plus x so for every time that i add x to 400 i'm subtracting x from the number of units that i sell and so these um i can then multiply these two binomials together and i end up with 44 000 110 times 400 plus 110 x 10 10 times x so i did these two here and then i have minus 400 x and then minus x squared so that's my revenue function and now i can combine like terms and i'm going to rearrange it so that my highest degree is at the beginning so i now have negative x squared i can combine these two right here and i have negative 290 x plus 44 000 so again we want to max maximize this...