Production For another product, the manufacturing firm in Exercise 31 estimates that production is a function of labor $x$ and capital $y$ as follows:

$$

f(x, y)=12 x^{3 / 4} y^{1 / 4}

$$

If $\$ 25,200$ is available for labor and capital, and if the firm's costs are $\$ 100$ and $\$ 180$ per unit, respectively, how many units of labor and capital will give maximum production?

## Discussion

## Video Transcript

all right, so we have an optimization problem and which we have a production formula in. Our goal is to maximize the production when the cost of the production can be no more than $40,000. So when we want to do is talk about our variables and talk about what we need to maximize staff, our cost function, so are variables. Are Axum y where X represents units of labor and why represents units of capital we need to create for this a cost function. So we're told that the, um, the problem is $80 a unit of labor. So if it is $80 per unit of flavor, then it is 80 x plus. We're told that is $150 per unit of capital. So that's 150. Why? And then again, we're told that we have $40,000. So our firm Ken's been no more than $40,000 at 80 x plus 150. Why? What we're going to do then, is take that cost function and we are going to get the problems of one variable using are constructed. So I'm gonna take the 40,000 equal to 80 x plus 150. Why? And we're going to solve that problem for why or for X Now, in this case, I chose to solve for X, but you could also solve her way. So I am going to subtract the 150 wife from both sides and divide by 80 and just a quick, simple simplifying of that. We get 500 minus 15 8th Why and then what I'm going to do is I'm going to substitute this piece of information in here in do my f of X by that I was given to start. So we're going to replace the X with that red function, so that now creates three and in terms off, why 500 minus 15? It's why, to the 1/3 power times wide of the 2/3. Why did I put this in terms of one variable? Well, being able to put it in terms of one variable allows me to find my derivative without needing to use implicit differentiation. It also allows me to find it derivative and find the zeros of that derivative by being able to find the zeros. I'll be able to find the why values at which the point could potentially be maximized. So what we're gonna do is we're gonna take our new function. We're going to find its derivative. So after prime, and in order to do that, we need to use the product room because we have two things raised to a power and we're actually going to use the chain role within the product roll. So the product rule says you have the first. Now we're gonna do is take it and multiply it by the derivative of the second. So the derivative of wider the 2/3 is 2/3. Why to the negative 1/3 and I were going toe. Add that to the second, which was, why did the 2/3 and multiply it by the derivative of the first? So that's three times 1/3. Keep the inside. Now we have to subtract one from the exponents on the outside, so you get negative 2/3 and then we're going to multiply that by the derivative of the insides, a derivative of 500 cancels and then derivative of negative 15 8th Why is negative? 15 8th now? We're just gonna do some simple cleaning up three times, 1/3 cancel three and 2/3. The threes are going to cancel, and then we are going to simplify this. So that gives me to times the quantity 500 minus 15 8th supply to the 1/3 all ever. Why? To the 1/3 minus, We're gonna pull the fraction out and front 15 8th and then we have why to 2/3. And that is going to be all over 500 minus 15 pates. Why? To the 2/3. In order to make this problem easier, we're gonna get a common denominator between the two. So that means that we need to multiply both fractions by the other denominators. So over here, two times eight is 16. 1/3 time's 2/3 will give us one. So we get 500 minus 15 eights. Why minus? And then if you multiply over here, why did the 2/3 times why do the 1/3 gives you why CF minus 15? Why all over this new denominator of eight to the 1/3 500 minus 15 8th Why? To the 2/3. And there you have our derivatives of our far are crossed production function sort of combined. So what we can do is use this to now find the points of which production could be maximized. So maximization can occur at the zeros of the derivative. So are gonna dio is set that derivative equal to zero. So in doing so, we get zero equals 16 times 500 minus 15 8th Why bias 15? Why now? We don't even worry about setting the denominator equal to zero because that will give me the points at which the derivative does not exist, which are vertical Assam tubes which would not be points of maximums or minimums. They wouldn't be an extra Emma. So what I did at this point was distributed thesixties een in. So I multiplied 16 by 500 and that gave me 8000. And then I multiplied 16 by 15/8 and what was nice about 16 times eight. Was it divided evenly and give you two. So you got negative 30 y minus 15. Why? And then at this point, we were able to get that are wise combined, say about 8000 minus 45. Why? And then I'm able to solve for why and I get the why is equal to 177.7 repeating. Okay, So if we want to use this line short to determine that this is a maximum, what will we have to do is put the 1177.7 on or sign shirt. We're using our F prime equation to determine this, and what you would do is just kind of plug in numbers to have prime to determine that it is positive to the left and negative to the right, meaning that the original would be increasing to decreasing. So again, if you just kind of quickly plugged in numbers to your original equation of the derivative and I actually plugged my numbers in to this form right here, so I'd say 8000 minus. And then again, you can pick kinda any number less than that so you could do 8000 minus 45 times 100 and we know that 8000 minus 45 times 100 would be 3500 which would give us a positive number. And then if you did this same and you pick a number larger than 1 77 So I picked 200. We can see easily that 8000 and 45 times 200 is going to give me in negative 1000. So there's the negative. And that means that the original is increasing, then decreasing. Which proves that this is the max value for why so then what we're gonna do to find X is plugged that value back in. So we have that X is equal to 500 minus 15 AIDS. Why? And that was from our simplified cost function and that we're just going to say OK, 500 minus 15 8 times 1 77.7 and that gives me 166.7. So we obviously cannot have 0.7 of a product. So what that means is that 177 units of capital and 166 units of labor well a maximize the company's production at a cost less than or equal to 40,000. And again, if you wanted to check your work and make sure your numbers were accurate, you could find the cost. At 1 66 1 77 by just plugging it in. So it would be 80 times when 66 plus 1 50 times 1 77 And you get that your costs is 39,000, 830. So we didn't.

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