You are given a pair of equations, one representing a supply curve and the other representing a demand curve, where p is the unit price for x items.
80 p + x - 380 = 0 and
84 p - x - 40 = 0
a. Identify which is the supply curve and demand curve and the appropriate domain. Put the domains in interval notation . For \infty type infinity . For more than one interval use a U to represent a "union".
Domain of the supply curve
Domain of the demand curve
b. Determine the market equilibrium. Equilibrium: x = p =
c. Determine the revenue function. Revenue function R(x)=
d. Determine the revenue at market equilibrium.
a. To identify which is the supply curve and demand curve, we can compare the coefficients of the price (p) and quantity (x) terms.
The equation 80p + x - 380 = 0 has a positive coefficient (80) for the price term (p), indicating that it represents the supply curve. The equation 84p - x - 40 = 0 has a negative coefficient (-1) for the quantity term (x), indicating that it represents the demand curve.
To find the domain of each curve, we need to determine the values of p and x that satisfy each equation.
For the supply curve, 80p + x - 380 = 0, we can solve for x:
x = 380 - 80p
The domain of the supply curve is typically non-negative values of x since quantities cannot be negative. Thus, the domain of the supply curve is x ≥ 0.
For the demand curve, 84p - x - 40 = 0, we can solve for x:
x = 84p - 40
Similar to the supply curve, the domain of the demand curve is typically non-negative values of x. Thus, the domain of the demand curve is also x ≥ 0.
Therefore, the domain of the supply curve and the demand curve is x ≥ 0.
b. To find the market equilibrium, we need to find the price (p) and quantity (x) at which the supply and demand curves intersect.
We can solve the given pair of equations simultaneously:
80p + x - 380 = 0 ---(1)
84p - x - 40 = 0 ---(2)
By adding equation (1) and equation (2), we eliminate the x term:
(80p + x - 380) + (84p - x - 40) = 0
164p - 420 = 0
Solving for p:
164p = 420
p = 420/164
p ≈ 2.56 (rounded to two decimal places)
Substituting the value of p back into equation (1), we can find x:
80(2.56) + x - 380 = 0
204.8 + x - 380 = 0
x = 380 - 204.8
x ≈ 175.2 (rounded to one decimal place)
Therefore, at market equilibrium, x ≈ 175.2 and p ≈ 2.56.
c. The revenue function (R) represents the total revenue generated by selling a certain quantity (x) at the given price (p). It is calculated by multiplying the price (p) by the quantity (x).
The revenue function is given by:
R(x) = p * x
Substituting the market equilibrium values, we get:
R(x) = 2.56 * 175.2
d. To determine the revenue at market equilibrium, we can substitute the market equilibrium values into the revenue function:
R(at market equilibrium) = 2.56 * 175.2
Calculating this value will give you the revenue at market equilibrium.