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.
A: Demand equation:
In order to find the demand equation, we can use the two points given: (250, 140) and (1000, 110).
First, let's calculate the slope (m) using the formula: m = (change in y) / (change in x).
m = (110 - 140) / (1000 - 250)
m = -30 / 750
m = -1/25
Now, let's use the point-slope form of a linear equation: y - y1 = m(x - x1), where (x1, y1) is one of the given points.
Using the point (250, 140):
y - 140 = (-1/25)(x - 250)
y - 140 = (-1/25)x + 10
y = (-1/25)x + 150
Therefore, the demand equation is y = (-1/25)x + 150.
B: Supply equation:
We are given two price-quantity combinations: (60, 750) and (80, 2250).
Using the same method, we can find the slope (m):
m = (2250 - 750) / (80 - 60)
m = 1500 / 20
m = 75
Using the point (60, 750):
y - 750 = 75(x - 60)
y - 750 = 75x - 4500
y = 75x - 3750
Therefore, the supply equation is y = 75x - 3750.
C: Equilibrium quantity and price:
To find the equilibrium quantity and price, we need to find the point where the demand and supply equations intersect. Set the demand and supply equations equal to each other:
(-1/25)x + 150 = 75x - 3750
Now, solve for x:
(75x + 1/25)x = 3750 + 150
(75 + 1/25)x = 3900
(1876/25)x = 3900
x = (3900 * 25) / 1876
x ≈ 51.91
Plugging this value back into either the demand or supply equation, we can find the equilibrium price:
y = 75(51.91) - 3750
y ≈ 2594.25
Therefore, the equilibrium quantity is approximately 51.91 units and the equilibrium price is approximately $2594.25.
To find the demand equation, we can use the points (quantity, price) given in the question.
Point 1: (250, 140)
Point 2: (1000, 110)
Let's use the slope-intercept form of a linear equation, y = mx + b, where y represents the quantity and x represents the price.
Step 1: Find the slope (m):
The slope (m) can be calculated using the formula: m = (y2 - y1) / (x2 - x1). Using the points above, we can plug in the values to find the slope:
m = (110 - 140) / (1000 - 250) = -0.03
Step 2: Find the y-intercept (b):
Using the slope-intercept form, we can rearrange the equation to solve for b:
b = y - mx
Let's use point 1 to calculate the y-intercept:
b = 140 - (-0.03 * 250) = 147.5
Therefore, the demand equation is:
Quantity demanded = -0.03 * Price + 147.5
Now let's find the supply equation.
Point 1: (750, 60)
Point 2: (2250, 80)
Again, let's use the slope-intercept form, y = mx + b, where y represents the quantity and x represents the price.
Step 1: Find the slope (m):
Using the formula, m = (y2 - y1) / (x2 - x1), we can calculate the slope using the points above:
m = (80 - 60) / (2250 - 750) = 1/150
Step 2: Find the y-intercept (b):
Using the slope-intercept form, we can rearrange the equation to solve for b:
b = y - mx
Let's use point 1 to calculate the y-intercept:
b = 60 - (1/150 * 750) = 55
Therefore, the supply equation is:
Quantity supplied = (1/150) * Price + 55
Finally, let's find the equilibrium quantity and equilibrium price.
Equilibrium quantity occurs when the quantity demanded equals the quantity supplied. We can set the demand equation equal to the supply equation and solve for the price:
-0.03 * Price + 147.5 = (1/150) * Price + 55
Simplifying the equation:
-0.03 * Price - (1/150) * Price = 55 - 147.5
-0.03 * Price - (1/150) * Price = -92.5
(-1/33.33) * Price - (1/150) * Price = -92.5
(-150 - 33.33) * Price = -92.5
(-183.33) * Price = -92.5
Price = -92.5 / -183.33
Price = 0.504
Now we can substitute this price back into either the demand or supply equation to find the equilibrium quantity. Let's use the demand equation:
Quantity demanded = -0.03 * 0.504 + 147.5
Quantity demanded = 142.89
Therefore, the equilibrium quantity is approximately 143, and the equilibrium price is approximately $0.50.
A: To find the demand equation, we can use the two data points given: when the unit price is $140, the quantity demanded is 250, and when the unit price is $110, the quantity demanded is 1000.
First, let's denote the unit price as "P" and the quantity demanded as "Q".
Using the two data points, we can create a system of equations:
Equation 1: 250 = a(140) + b
Equation 2: 1000 = a(110) + b
To solve this system, we can use the method of substitution.
From Equation 1:
250 = 140a + b
Solving for b, we get:
b = 250 - 140a
Substituting this value of b into Equation 2, we get:
1000 = a(110) + (250 - 140a)
Simplifying, we have:
1000 = 110a + 250 - 140a
1000 - 250 = -30a
750 = -30a
a = -25
Substituting this value of a back into Equation 1, we get:
250 = -25(140) + b
250 = -3500 + b
250 + 3500 = b
b = 3750
Therefore, the demand equation is:
Q = -25P + 3750
B: To find the supply equation, we are given that the suppliers will market 750 espresso makers if the unit price is $60 or higher, and they will market 2250 units at a unit price of $80.
Using the same notation as before (P for unit price and Q for quantity supplied), we can create a linear equation using the two data points given:
Equation 3: 750 = c(60) + d
Equation 4: 2250 = c(80) + d
Similarly, using the method of substitution, we can solve this system.
From Equation 3:
750 = 60c + d
Solving for d, we get:
d = 750 - 60c
Substituting this value of d into Equation 4, we have:
2250 = c(80) + (750 - 60c)
Simplifying:
2250 = 80c + 750 - 60c
2250 - 750 = 20c
1500 = 20c
c = 75
Substituting this value of c back into Equation 3, we get:
750 = 75(60) + d
750 = 4500 + d
750 - 4500 = d
d = -3750
Therefore, the supply equation is:
Q = 75P - 3750
C: The equilibrium quantity and price occur when the quantity demanded equals the quantity supplied. To find this, we set the demand equation equal to the supply equation:
-25P + 3750 = 75P - 3750
Adding 25P to both sides:
3750 = 100P - 3750
Adding 3750 to both sides:
7500 = 100P
Dividing by 100:
P = 75
Substituting this value of P back into either the demand or supply equation, we get:
Q = -25(75) + 3750
Q = -1875 + 3750
Q = 1875
Therefore, the equilibrium quantity is 1875 and the equilibrium price is $75.
demand:
(d-250)/(p-140) = (1000-250)/(110-140)
d-250 = -25(p-140)
d = -25p + 3750
supply:
(s-750)/(p-60) = (2250-750)/(80-60)
s-750 = 75(p-60)
s = 75p - 3750
equilibrium:
-25p + 3750 = 75p - 3750
100p = 7500
p = 75
s = d = 1875