Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

Suppose it costs $\$ 8$ to produce each unit of Product 1 and $\$ 12$ to produce each unit of Product $2 . x$ units of Product 1 and $y$ units of Product 2 are produced, and the fixed overhead cost is $\$ 10 .$ Suppose the two products have demand equations $p_{1}=-2 x+y+20$ and $p_{2}=3 x-5 y+12,$ where $p_{1}$ and $p_{2}$ are the per unit prices in dollars. Find the demand levels for the two products if profit it to be maximized.

$$(5,2)$$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 5

Economic Applications

Partial Derivatives

Johns Hopkins University

Missouri State University

Campbell University

University of Michigan - Ann Arbor

Lectures

12:15

In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

04:14

02:07

The demand equation for a …

06:00

Maximizing Profit A compan…

05:42

Maximizing Profit A monopo…

07:41

Two products are manufactu…

05:14

Profit from Two Products A…

today and today we have a question on maximizing the profits. Uh Mhm. Equation. So we are given the cost function for a certain to products and we are given how much they sell for. And they were also given a little useful hint that profit equals revenue minus costs for in case you have gotten that. So how do we go about this question? Well we the since we're trying to maximize products, we should probably first go and try and get and some sort of formula for profits. So this profit equals revenue minus costs. So what is the revenue? We know that products excels for $10 in product white cells, food uh $9. So then revenue simply 10 X plus nine Y. X. Being the amount of sold of one product and why being that of the other? And then that we simply minus the cost function. So there we go. And let's quickly uh Mhm. Just simplify that and there we go. You see I went ahead and signed this now to a new function P in the point. Xy. So that's just P for profits. It doesn't really matter what we call it. Uh And then to maximize this we of course go find the first derivatives in terms of X and Y. So derivative of P. In terms of X gives us negative 0.6 X squared just X plus eight X. Or just eight plus eight or are delicious. Go first minus 0.1 Why plus eight? And then we find the relative P. In terms of Y. And that is also negative 0.6 Y minus 0.1 X plus six. Yeah. And then we know that if we're trying to maximize or minimize something, then we need to find where the first two relatives are equal to zero. So that means that we end up with two equations. And this is simply solving two equations for two variables. So let's first go right X. In terms of why we can also do this with major matrices. But since we're doing calculus and algebra others, let's just do it on the old fashioned the old fashioned way. So we go 0.6 X minus 0.1 Y plus eight equals to zero. Then that comes to this, which simplifies to this. Over here. Now we have three equations. Let's just go ahead and number them. We have this equation number one, question number two, and then this would be equation number three, which was derived from equation number one. So then if we go ahead and we substitute equation number three into equation number two comes out to this. And then if you go ahead and simplify it, we get and that comes out to this, you'll notice that I changed a bit something up here since I made a small copying error. But then if we simplify this further, we get this and then finally we get y is equal to 80 units mm And then X of course to get that, we just go for all this new information information and we'll take this new information and it's just that is for just for the sake of it. And then you guys substitute that into equation three X is equal to negative one of six types 80 plus 403. Quite a common village said myself work we have to do, but then this is equal to 120 units. And so what exactly did they find us to find the values of X and Y. That maximize the company's profits? So other words, profits are maximized at x equal to 120 units and why equal to 80 units. And there we go. That's the question answered

View More Answers From This Book

Find Another Textbook

03:05

Use the method of Lagrange multipliers to optimize $f$ as indicated, subject…

01:21

Linearize the given function.$$f(x, y)=x^{4}+y^{4}+16 x y \text { near }…

01:39

Find and classify, using the second partial derivative test, the critical po…

01:32

Evaluate the given integral and check your answer.$$\int 3 d x$$

05:00

Find the partial derivatives with respect to (a) $x,$ (b) $y$ and (c) $z$.

01:01

Solve the given differential equation.$$\frac{d y}{d x}=\frac{4 e^{x}}{y…

01:05

If $f^{\prime}(x)=\frac{-2}{x^{2}},$ sketch the family of solutions.

05:27

Find the partial derivatives with respect to (a) $x$ and (b) $y$.$$f(x, …

04:04

Evaluate the given integral.$$\int \frac{(\ln x)^{N}}{x} d x$$

01:29

Evaluate the given definite integral.$\int_{4}^{9} \sqrt{t} d t$