Mathematics, 19.11.2019 08:31 laladance123

The quantity demanded of a certain brand of dvd player is 3000/wk when the unit price is $485. for each decrease in unit price of $40 below $485, the quantity demanded increases by 500 units. the suppliers will not market any dvd players if the unit price is $334 or lower. but at a unit price of $559, they are willing to make available 2500 units in the market. the supply equation is also known to be linear.

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Answer from: Star6147

Given that the quantity demanded of a certain brand of DVD player is 3000/wk when the unit price is $485. For each decrease in unit price of $40 below $485, the quantity demanded increases by 500 units. The suppliers will not market any DVD players if the unit price is $334 or lower. But at a unit price of $559, they are willing to make available 2500 units in the market. The supply equation is also known to be linear.

Part (a): Find the demand equation.
To find the demand equation, we notice that the demand function is linear because we were told that for each decrease in unit price of $40 below $485, the quantity demanded increases by 500 units.

Given that at a price of $485, the quantity demanded is 3,000 and given that for each decrease in unit price of $40 below $485, the quantity demanded increases by 500 units. Then for a price of $485 - $40 = $445, the quantity demanded is 3,000 + 500 = 3,500 units

Thus, using the formula for the equation of a straight line and the points (3,000, 485) and (3,500, 445), we find the demand equation as follows:

$\frac{p-485}{q-3,000} = \frac{445-485}{3,500-3,000} \\ \\ \Rightarrow \frac{p-485}{q-3,000}= \frac{-40}{500} =- \frac{2}{25} \\ \\ \Rightarrow25(p-485)=-2(q-3,000) \\ \\ \Rightarrow25p-12,125=-2q+6,000 \\ \\ \Rightarrow25p+2q=18,125$

Part (b): Find the supply equation.
To find the supply equation, given that the supply function is linear and that the suppliers will not market any DVD players if the unit price is $334 or lower. But at a unit price of $559, they are willing to make available 2500 units in the market.

Thus, using the formula for the equation of a straight line and the points (0, 334) and (2,500, 559), we find the demand equation as follows:

$
 \frac{p-334}{q-0} = \frac{559-334}{2,500-0} \\ \\ \Rightarrow
 \frac{p-334}{q}= \frac{225}{2,500} =\frac{9}{100} \\ \\ 
\Rightarrow100(p-334)=9q \\ \\ \Rightarrow100p-33,400=9q
\\ \\ \Rightarrow100p-9q=33,400$

(c) Find the equilibrium quantity and price.
The equilibruim quantity and price is given by the quantity and price at which the quantity demanded is equal to the quantity supplied. It is obtained by equating the demand and the supply equations.

i.e.
From demand equation, we have
$25p+2q=18,125 \\ \\ \Rightarrow25p=-2q+18,125 \\ \\ \Rightarrow p= -\frac{2}{25}q +725 \ . \ . \ . (1)$

From supply equation, we have
$100p-9q=33,400 \\ \\ \Rightarrow100p=9q+33,400 \\ \\ \Rightarrow p= \frac{9}{100} q+334 \ . \ . \ . (2)$

Equating equations (1) and (2), we have
$-\frac{2}{25}q +725=\frac{9}{100} q+334 \\ \\ \Rightarrow\frac{9}{100} q+\frac{2}{25}q=725-334 \\ \\ \Rightarrow \frac{17}{100} q=391 \\ \\ \Rightarrow q= \frac{391\times100}{17} =2,300$

Substituting for q into equation (1), we have
$-\frac{2}{25}(2,300) +725=-184+725=541$

Therefore, the equilibruim quantity is 2,300 units at a price of $541.

Answer from: Quest

answer: a video game has three difficulty levels: easy, medium, and expert. the game's scoring works in such a way that a player has to score a fixed number of points to advance to the next level, and then the scoring starts over. the total score needed to clear all three difficulty levels is 7,500. two times the number of points needed to clear the easy level is 500 more than the number of points needed to clear the medium level. twice the number of points needed to clear the medium difficulty level is 1,500 more than the number of points needed to clear the expert level. if x, y, and z represent the easy, medium, and expert levels, respectively, and a is the coefficient matrix of the system of equations modeling this situation, identify the inverse matrix, a-1, and the solution matrix, x, of the system of equations.

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Answer from: Quest

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