SM2 17.4.4 Finding Revenue Function and Maximizing Profit

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[00:00:01]Hello and welcome. So the cost in dollars to  produce some number of designer dog leashes is C of x equals 18x plus 8 and the number of  leashes sold is N of x equals 98 minus x. So when we're looking at this it says find the revenue  function. All right. So when we're looking at this you'd say all right well revenue. Revenue  is essentially the amount of money you get from the price of the dog leashes times the number  that you sell. So in other words we have the amount we're selling the dog leashes for times  our number of leashes being sold, right? We're not looking for profit we're just looking for straight  revenue. So all we need to do is multiply these two functions and there we go. So we're going to  start with our C of x: 18x + 8 times our N of x: 98 minus x. Now these are going to be some large  numbers because our first goal is then distribute to multiply these two together. Now keep in mind  the order didn't matter. We could have put those parentheses in any order. The important thing is  that we're multiplying them all right. So first times first we have 18 times 98. I would honestly  just get a calculator for that and it's 1,764 x.

[00:01:25]And then the first times the second so 18x times  negative x is negative 18 x squared. And then we go the other direction where now it's going to  be 8 times 98 so 784 and then 8 times negative x so negative 8x. From there we would just combine  like terms to simplify and there is your revenue function. So you only have one x squared term:  negative 18 x squared. That will go first. Combine your x terms together so we have 1,764 minus  8 which is just going to give us plus 1,756 x, right? And then the last but not least we have our  just constant, right? And there's nothing that's going to combine with that so we would just add  that there at the end: 784. Right? Now the next question is going to ask us to find the number of leashes  which need to be sold to maximize our revenue.

[00:02:28]So hopefully you know that anything with a square  on there is going to imply that it's a quadratic.

[00:02:36]Quadratic is a U-shaped graph that could go  up or it could go down. So the first thing you look at is the direction of opening. What is this  quadratic doing? Well the quadratic you can look from your leading coefficient is negative 18 which  means the quadratic the arrows will be facing downwards. So more or less on our graph it's  going to be doing something kind of like this, right? More or less where it's going to be facing  downwards. And when we're talking maximizing our profit we're going to be talking about the x  number of leashes and our y our y axis is going to be your money. So for the number of leashes  that you sell you want to make the most money and we're essentially trying to find where is that  most money going to be made. In other words the vertex of our quadratic is what we're looking for.  So we want to find the highest point. There's a couple ways to do that. You could find your two x  intercepts, add them together, divide by two — in other words take the average. But hopefully, you  remember that there is a shortcut when it comes to finding your vertex for standard form and that  is negative b divided 2a and that will find you the x value of your vertex, right? Your h value. And  that'll be your first number and that's what this first blank is asking us for. Okay. So let's go  ahead and do that. The a, b, and c. The a is your number at the very beginning so negative 18. B is  our number in the middle tied to our x so that's 1,756. So let's just calculator this. So we have  — let me plug that in — 1,756 divided by 2 times negative 18. Oh and don't forget it's  negative b there we go. All right and we get 48.7... repeating. Now when we're looking at  this we're thinking about dog leashes. You're not going to sell 0.7 of a dog leash or  even if your decimal was lower. If that was 0.1 of a dog leash you would still need to sell more  than 48 dog leashes. We have a decimal there. You would actually round up regardless of the number  you're looking at. So we have 48.7 repeating which means we'll need 49 leashes in order to  maximize our profit, right? We're rounding up to the nearest whole leash. If your decimal  was like 48.1 you would still turn that into 49 leashes because you need 0.1 of a leash which  means you just need another leash, right? So keep that in mind. Regardless of your decimal round  that answer up. So you need 49 leashes. Okay.

[00:05:26]Now we want to know what the maximum revenue is.  In other words we want to know the dollar signs.

[00:05:31]If we sell these 49 leashes how much money are we  going to make kind of an idea. All right. So to do that we need to find the y value of our graph and  remember this is a function meaning R of x that is your y value. So if you plug in the x you find  the y and we just happen to know the x value is 49 which means that that's how we're going to find  y value. You're not going to put the 48.7 in here because you need 49 leashes, right? You can't make  0.1 or 0.9 or 0.7 of a leash. You need a whole leash. So we're going to put 49 in. All right. So  negative 18 times 49 squared plus 1,756 times 49 times - I have to move this over... or, apparently not... there- there we go.

[00:06:31]um plus 784. That is your goal. All right. So you  just plug this right into a calculator. So we have negative 18 times 49 squared and then  plus, uh... I forgot the number already, plus 1,756 times 49 plus I believe it was 784. Yes it was.

[00:07:01]Okay perfect. So we get 43,610. So $43,610 is our  final answer. Um and that's going to be in money.

[00:07:15]That's going to be in dollars. Okay. We're finding  how much money we're making. That's quite a lot of money. Um designer dog leashes are apparently  fantastic for business. So there you go. All right and that's how you would solve all of these  and there you go. Thank you so much for watching.