Respuesta :
Answer:
(275,122)
When the price will be $275, the quantity will be 122 televisions.
Step-by-step explanation:
We have been given that a group of retailers will buy 104 televisions from a wholesaler if the price is $300 and 144 if the price is $250.
As the quantity of televisions depends on price of televisions, so our demand curve will pass through points (300,104) and (250,144).
Let us find slope of demand line using slope formula.
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex], where,
[tex]m =\text{Slope of line}[/tex],
[tex]y_2-y_1=\text{Difference between two y-coordinates}[/tex],
[tex]x_2-x_1=\text{Difference between same x-coordinates of two y-coordinates}[/tex]
Upon substituting coordinates of our given points in slope formula we will get,
[tex]m=\frac{104-144}{300-250}[/tex]
[tex]m=\frac{-40}{50}[/tex]
[tex]m=-\frac{4}{5}[/tex]
Let us substitute [tex]m=-\frac{4}{5}[/tex] coordinates of point (250,144) in slope intercept form of equation ([tex]y=mx+b[/tex]).
[tex]144=-\frac{4}{5}*250+b[/tex]
[tex]144=-4*50+b[/tex]
[tex]144=-200+b[/tex]
[tex]144+200=-200+200+b[/tex]
[tex]344=b[/tex]
Upon substituting [tex]b=344[/tex] and [tex]m=-\frac{4}{5}[/tex] we will get equation of our demand line as:
[tex]y=-\frac{4}{5}x+344[/tex]
Similarly we will find the equation of supply line using points (225,88) and (315,168).
[tex]m=\frac{168-88}{315-225}[/tex]
[tex]m=\frac{80}{90}[/tex]
[tex]m=\frac{8}{9}[/tex]
Let us substitute [tex]m=\frac{8}{9}[/tex] and coordinates of point (225,88) in slope intercept form of equation ([tex]y=mx+b[/tex]).
[tex]88=\frac{8}{9}*225+b[/tex]
[tex]88=8*25+b[/tex]
[tex]88=200+b[/tex]
[tex]88-200=200-200+b[/tex]
[tex]-112=b[/tex]
Upon substituting [tex]b=-112[/tex] and [tex]m=\frac{8}{9}[/tex] we will get equation of our supply line as:
[tex]y=\frac{8}{9}x-122[/tex]
Let us equate both lines to find the point where both lines intersect.
[tex]-\frac{4}{5}x+344=\frac{8}{9}x-122[/tex]
[tex]-\frac{4}{5}x-\frac{8}{9}x+344=\frac{8}{9}x-\frac{8}{9}x-122[/tex]
[tex]-\frac{4}{5}x-\frac{8}{9}x+344=-122[/tex]
[tex]-\frac{4}{5}x-\frac{8}{9}x+344-344=-122-344[/tex]
[tex]-\frac{4}{5}x-\frac{8}{9}x=-466[/tex]
Let us have a common denominator.
[tex]-\frac{4*9}{5*9}x-\frac{8*5}{9*5}x=-466[/tex]
[tex]-\frac{36}{45}x-\frac{40}{45}x=-466[/tex]
[tex]\frac{-36-40}{45}x=-466[/tex]
[tex]\frac{-76}{45}x=-466[/tex]
[tex]\frac{-76}{45}*\frac{45}{-76}x=-466*\frac{45}{-76}[/tex]
[tex]x=466*\frac{45}{76}[/tex]
[tex]x=6.1315789473684211*45[/tex]
[tex]x=275.9210526315789495\approx 275[/tex]
Let us substitute x=275 in any of our equation to solve for y.
[tex]y=-\frac{4}{5}*275+344[/tex]
[tex]y=-4*55+344[/tex]
[tex]y=-220+344[/tex]
[tex]y=122[/tex]
Therefore, the equilibrium point for the market will be (275,122).
The equilibrium point will be around 128 units, which will be supplied and demanded at an equilibrium price of $ 270.
Since a group of retailers will buy 104 televisions from a wholesaler if the price is $ 300 and 144 if the price is $ 250, while the wholesaler is willing to supply 88 if the price is $ 225 and 168 if the price is $ 315, assuming the resulting supply and demand functions are linear, to find the equilibrium point for the market the following calculation must be performed:
- (300 - 250) / (144 - 104) = X
- 50/40 = X
- 1.25 = X
- Thus, the retailers will buy one more television for every $ 1.25 of sale.
- (315 - 225) / (168 - 88) = X
- 90/80 = X
- 1.125 = X
- In turn, the wholesaler will supply one less television for every $ 1,125 of discount.
- 116 x (300 - 12 x 1.25) = X
- 116 x 285 = X
- 33060 = X
- 116 x (225 + 28 x 1.125) = X
- 116 x 256.5 = X
- 29754 = X
- 126 x (300 - 22 x 1.25) = X
- 126 x 272.5 = X
- 34335 = X
- 126 x (225 + 38 x 1.125) = X
- 126 x 267.75 = X
- 33736.5 = X
- 128 x (300 - 24 x 1.25) = X
- 128 x 270 = X
- 34560 = X
- 128 x (225 + 40 x 1.125) = X
- 128 x 270 = X
- 34560 = X
Therefore, the equilibrium point will be around 128 units, which will be supplied and demanded at an equilibrium price of $ 270.
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