The quantity demanded each month of russo Espresso Makers is 250 when the unit price is $140; the quantity demanded each month is 1000 when th e unit price is $110. the suppliers will market 750 expresso makers if the unit price is $60 or higher. At a unit price of $80 they are willing to market 2250 units Both the demand and supply equations are known to be liniear.
A: Find the demand equation.
B: Find the supply equation.
c. find the equilibrium quantity and the equilibrium price.
45
y=-25p+3750
A: To find the demand equation, we need to determine the slope (m) and the y-intercept (b) of the demand line.
Given information:
When the unit price is $140, the quantity demanded is 250.
When the unit price is $110, the quantity demanded is 1000.
Using the formula for the slope of a line (m = Δy/Δx), we can calculate the slope:
m = (1000-250) / ($110-$140)
= 750 / (-30)
= -25
Now, we can use the point-slope form of a linear equation (y - y1 = m(x - x1)) and substitute the coordinates of one of the given points (e.g., $140, 250):
y - 250 = -25(x - $140)
Simplifying:
y - 250 = -25x + $3500
Rearranging to the standard form:
25x + y = $3750
So, the demand equation is 25x + y = $3750.
B: To find the supply equation, we need to determine the slope (m) and the y-intercept (b) of the supply line.
Given information:
When the unit price is $60 or higher, the quantity supplied is 750.
When the unit price is $80, the quantity supplied is 2250.
Using the formula for the slope of a line (m = Δy/Δx), we can calculate the slope:
m = (2250-750) / ($80-$60)
= 1500 / 20
= 75
Now, we can use the point-slope form of a linear equation and substitute the coordinates of one of the given points (e.g., $60, 750):
y - 750 = 75(x - $60)
Simplifying:
y - 750 = 75x - $4500
Rearranging to the standard form:
-75x + y = -$3750
So, the supply equation is -75x + y = -$3750.
C: To find the equilibrium quantity and price, we need to find the point where the demand and supply equations intersect.
We can solve the system of equations by setting the demand equation equal to the supply equation:
25x + y = $3750
-75x + y = -$3750
By rearranging the second equation, we can express y in terms of x:
y = 75x - $3750
Now, substitute this value of y into the first equation:
25x + (75x - $3750) = $3750
Simplify:
100x - $3750 = $3750
Add $3750 to both sides:
100x = $7500
Divide both sides by 100:
x = $75
Substitute this value of x back into either equation:
y = 75x - $3750
y = 75($75) - $3750
y = $5625 - $3750
y = $1875
Therefore, the equilibrium quantity is 1875 and the equilibrium price is $75.
To find the demand equation, we can use the given information about quantity demanded at different price levels.
Let's assign the variables as follows:
- Quantity Demanded (D) as Q
- Unit Price (P) as p
We have two data points:
1. Q = 250 when p = 140
2. Q = 1000 when p = 110
Using the slope-intercept form of a linear equation (y = mx + b), we can calculate the equation for demand.
1. Calculate the slope (m):
m = (Q2 - Q1) / (p2 - p1)
= (1000 - 250) / (110 - 140)
= 750 / -30
= -25
2. Substitute one set of values from the data points, let's use the second set:
p = 110, Q = 1000
1000 = -25 * 110 + b
1000 = -2750 + b
b = 1000 + 2750
b = 3750
Therefore, the demand equation is:
D = -25p + 3750
To find the supply equation, we can use the given information about the quantity supplied at different price levels.
We have two data points:
1. Q = 750 when p = 60
2. Q = 2250 when p = 80
Using the same approach as before, we'll calculate the supply equation.
1. Calculate the slope (m):
m = (Q2 - Q1) / (p2 - p1)
= (2250 - 750) / (80 - 60)
= 1500 / 20
= 75
2. Substitute one set of values from the data points, let's use the second set:
p = 80, Q = 2250
2250 = 75 * 80 + b
2250 = 6000 + b
b = 2250 - 6000
b = -3750
Thus, the supply equation is:
S = 75p - 3750
To find the equilibrium quantity and equilibrium price, we need to set the demand and supply equations equal to each other:
D = S
-25p + 3750 = 75p - 3750
Rearrange the equation:
100p = 7500
p = 75
Therefore, the equilibrium price is $75.
Substitute this value back into either the demand or supply equation to find the equilibrium quantity.
Using the demand equation:
D = -25 * $75 + 3750
D = -1875 + 3750
D = 1875
Therefore, the equilibrium quantity is 1875 units.
To summarize:
- The demand equation is D = -25p + 3750.
- The supply equation is S = 75p - 3750.
- The equilibrium price is $75.
- The equilibrium quantity is 1875 units.