Supply and Demand problem

A certain company offers two products (a, b, measured in thousands of units). Their respective demand equations are: \(4a+p_1=14\) and \(2b + p_2 = 24\) where \(p_1\), \(p_2\) stand for respectively- the selling price per unit of each product. The overall cost is given by \(a^ 2 + 5ab + b ^ 2\) Study the maximum profit. (2.5 points)

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To study the maximum profit for the given company, we need to:

1. Define the profit function

a, b: quantities sold (in thousands)

\(p_1, p_2\): prices per unit of products a and b

2. From demand equation s, solve for price as a function of quantity:

Given:

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ترسیم با معادلات ریاضی

معادلات مورد نیاز

$$\pm{{\sqrt{4-(x-2)^2 \over 3}}}$$

$$2 \pm \sqrt{-3x^2+4}$$

$$\pm{{\sqrt{4-(x+2)^2 \over 3}}}$$

$$-2 \pm \sqrt{-3x^2+4}$$

𝑥^2−𝑥𝑦+𝑦^2−√2 𝑥−√2 𝑦=0

$${{x+\sqrt{2}}\pm \sqrt{-3x^2+6x\sqrt{2}+2}} \over2$$
𝑥^2+𝑥𝑦+𝑦^2+√2 𝑥−√2 𝑦=0

$${-{x-\sqrt{2}}\pm \sqrt{-3x^2+6x\sqrt{2}+2}} \over2$$
𝑥^2−𝑥𝑦+𝑦^2+√2 𝑥+√2 𝑦=0

$${{-x+\sqrt{2}}\pm \sqrt{-3x^2-6x\sqrt{2}+2}} \over2$$
𝑥^2+𝑥𝑦+𝑦^2−√2 𝑥+√2 𝑦=0

$${{x-\sqrt{2}}\pm \sqrt{-3x^2-6x\sqrt{2}+2}} \over2$$

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در GeoGebra ببینید (TeX)
$$sin(\pi ^3x)  {\sqrt{e^2-x^2\over 2} }+\sqrt{|x|}$$

Chapter 5.4 _ Longman Physics 11-14

Check this above image for a multiple choise quiz

1) First comases made up of _____________ called _____________.

2) What we call lodesstone now?

3) Name three feromagnetic materials?

4) Why steel has also magnetic properties?

5) In a magnet, which area is called magnetic pole?

6) What happen if you put two North poles together?

7) What will happen if a north pole and a south pole put together?

8) A piece of iron will be attracted to a magnet, why iron is not a magnet?

9) How can we know that two metals are magnets?

10) How we can magnetise a piece of iron?

11) Why a typical piece of iron is not a magnet?

12) What is a saturated magnet?

13) When we have a saturated magnet?

14) Write two types of magnetic material and give an example for each of them.

15) Why iron is a soft magnetic material?

16) Why steal is a hard magnetic material?