For this problem. We are looking at a machine shop that makes bolts. If you read through the problem, it's always good to read through the whole problem. Before you start trying to do something with this many numbers, you read through the problem. You can see that we're trying to maximize the revenue of this company. So this is a maximization problem. And what is my revenue? Well, my revenue is I make 15 cents for each bolt of type one, so I'm going to mark, except one. This is type one for my bolts, and I make a second bolt That gives me 20 cents in revenue. So we let, except to be type two. Okay, So if I knew how many of each type of bold I made, I could find my revenue again. This is incense. So at the very end will probably have to convert it to dollars. But this is my revenue. My constraints happen at my three machines. For machine one, I need to spend 0.2 minutes for each of bolt type one. And for bolt type two, I only have 300 minutes available. So all the time that I spend on. These has to be less than or equal to 300. Machine to I have a little more time. Actually have 720 minutes for machine too. Everyone of type one bolt takes 10.6 minutes. My type. Choose each take point to again. Now type three is the fastest. I only need 4/10 of a minute for type one bolts and 8/10 of a minute for type two bolts. And for that machine, I have 100 minutes each day that I can use. So these are my constraints on. I want to use them to find my maximum revenue. Now, I'm gonna be using my desk. Most graphing calculator to see what this looks like. You can use this graphing application, your your calculator, anything that you're comfortable with. And if I put these on here, these are three constraints. You can see I have a nice area where they overlap. Five boundary points are there. Now, My fifth boundary, 0.0 I am not even going to consider that. That means I have made no bolts, so I have no profit. That's my minimum profit. So certainly don't wanna look at that for maximizing profit. So I'm gonna look at the other four points. So if I take them and I'm not gonna write them no particular order 0, 1250 501,000, 1050 and 450 and 1200 0 those are the four boundary points I care about again. I'm not even looking at 00 So if I plug those points into my revenue function, I confined this with my first point. The revenue is 25,000 pennies or $250. My second point gives me a revenue of 27,500 the 3rd 1. 24,750. And the last one is 18,000 sets. If I'm looking to maximize, I'm going to find that right here. That maximum is going to be $275 and I'll need 500 bolts of type one and 1000 volts of type two. Now, what if I make some changes specifically, I'm going to change this number right here. My revenue for type one bolt. Currently it's 15 cents. But what if I start to increase it At what point will I increase it? Enough to change my answer. To change what? Uh, allocation of type one and type two bowls. I want to dio. Well, let's go back and look at our equation here right now. Going to remove thes. Okay, Right now that is our maximum profit at 501,000 if I increase the the costs are that the revenue for type one? That's going to mean that my slope is going to get larger and larger, and it's a negative number, So it's gonna be a larger and larger negative slope that's going to put me at that point. So that's gonna be my next If I do make a shift to a higher um uh, make a shift to a different allocation based on a higher revenue for part one, that's going to be my new allocation. So let's take a look at what I would have to have is my new price point. Well, I'm going to introduce two new variables here. I'm going to say, let me use why is just to keep them separate from my exes? Why one is going to be the revenue for type one. And why to is the revenue and the new for type two right now, those air 15 and 20. But I'm gonna be changing them. So right now, from my maximum profit, I have simply and I don't wanna lose that. We're just gonna keep that. That's for the first part. Right now, I have 500 units of type one, so my revenue is going to be 500 times the revenue for that type. That's my overall revenue. And I have 1000 of type two. That's my revenue. Currently, if I were to go to my next, uh, allocation point, that would mean I'd have a revenue off 1050 of type one and 450 of type two. Right now, that is true work, because this is this is my highest revenue at the moment, 500,000 that split. But I want to know at what point does it change? At what point will this side be higher Now, if I solve this, I'm going to subtract 450 from both sides are 450 wise up to, and I'm going to subtract 500 wise up one so I can simplify that it will happen when the revenue for type one is higher than the revenue for type two. Well, we know that the revenue for type two is 20. That is not changing. So I will change my my allocation point if my revenue for type one is higher than 20. And let's just kind of show that I'm gonna put a little bit. I'm only gonna look at these two points right here. I'm not gonna look at the top or the bottom. At my current 500 1000. Let's say I have a price point. Are the revenue point for that type one? Let's make it be 1999 20 which should be where they're equal and 20.1 Now I know these air pennies. I can't really have 1/100 of a penny, but I just want to see that little bit of a difference on either side. So where I currently am at 500 of Type one in 1000 of type two at 19.99 cents apiece, 29995 That is the max. That's my total revenue. At 20 it is 30,000, and at 20.1 it is 30,000 and five. What about this second point 1050 of type 1 450 of type two? If I look at that when it's 20 they are indeed equal. At 19.99 it has a revenue of 29 not 89.5. It is less and a 20.1 It is 30,000 10.5 I have not by much, but it is Mawr. So right there, that's where I'm going to change. If the revenue for Type one goes above 20 that's when we're going to switch to a new allocation.