14) A manufacturer produces two products, product A and...

14) A manufacturer produces two products, product A and product B. Both products require processing on Machines I and II. The number of hours needed to produce one unit is given by the following chart: | Machine I | Machine II Product A | 2 hrs | 3 hrs Product B | 1 hrs | 4 hrs Machine I is available for at most 1000 hours and Machine II is available for at most 2500 hours. If the profit made on product A is $20 / unit and the profit made on product B is $25 / unit. Find the production level that will maximize profit and find the Maximum profit.

Transcript

00:01 Alright, so we have this company and they produce, they have two products, product a and product b.

00:05 Product a has to work on machine one for two hours and machine two for three hours.

00:12 Product b, it's one hour on machine one and four hours on machine two.

00:19 You can only use machine one a thousand hours for the day or the week or whatever and you can use machine two, it's only available for 2 ,500 hours during the week or month or whatever the time unit is.

00:32 Product a gives us a profit of $20 per unit, product b gives us a profit of $25 per unit.

00:40 Okay, so we want to maximize our profit.

00:46 Alright, so here we go.

00:50 How do we do this? alright, so machine one, right, if we look at machine one, we know that if we go to a, right, two times product a plus one times product b, we know that that has to be less than a thousand hours, less than or equal to 1 ,000 hours because that's all the time we have for machine one.

01:31 If we do the same thing for machine two, three times a plus four times b has to be less than or equal to 2 ,500 because that's how much we're limited by the hours of machine b.

01:55 So if we graph these, if you think of a little graph here, here's product, we'll call this one product a down here and this will be product b up here along the side.

02:13 So when we graph that first equation, two a plus b, if we don't make any product a and we use all b, we're going to be up here at 1 ,000.

02:27 If we don't make any product b and we only use product a, make product a, this will be here at 500.

02:39 So this is our first little kind of constraint line and we can be less than that.

02:47 We can be inside of that.

02:49 Okay, if we look at machine two, i'll graph this one in red so you can tell.

02:55 If we don't make any product a's and we only make product b, 2 ,500 divided by four is 625.

03:10 So that's about right here.

03:12 If we only make product a and no product b, 833 .333, that's like right here, that's like right there.

03:31 So that's this constraint and we'd shade inside.

03:38 So our shades overlap in this region right here.

03:43 So one of the coordinates of the boundaries, one of those coordinates, either this one here, zero, zero, it's not going to be zero, zero.

03:54 This one here that we don't know, 500 comma zero or zero comma 625.

04:02 One of those is going to maximize our profit.

04:05 It's probably going to be this guy right here, but we don't know what that one is.

04:10 But to find it, what you do is take the top equation.

04:14 I'm going to multiply by negative four.

04:17 So we have negative 8a minus 4b.

04:21 I'm going to just say equal to negative 4 ,000.

04:26 And then the other equation is 3a plus 4b equals 2 ,500.

04:35 And when we add straight down, the b's are going to disappear.

04:39 We have negative 5a equals negative 1 ,500.

04:45 And when we divide by five, a is going to be, what is that, 300 right there.

04:54 So this point is 300 comma...

Question

Please give Ace some feedback