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.
To identify the supply and demand curves and determine their domains, we need to rearrange the equations in terms of x.
The first equation is:
80p + x - 380 = 0
We can rearrange this equation to:
x = 380 - 80p
The second equation is:
84p - x - 40 = 0
Rearranging, we get:
x = 84p - 40
a. The equation x = 380 - 80p represents the supply curve since it shows the quantity (x) supplied at different prices (p). The equation x = 84p - 40 represents the demand curve as it shows the quantity (x) demanded at different prices (p).
To determine the domain, we need to consider the possible values of p that make the equations valid. Since quantity (x) cannot be negative in this context, we set x ≥ 0.
For the supply curve, x = 380 - 80p, we solve for p:
380 - 80p ≥ 0
-80p ≥ -380
p ≤ 4.75
So the domain of the supply curve is (-∞, 4.75].
For the demand curve, x = 84p - 40, we solve for p:
84p - 40 ≥ 0
84p ≥ 40
p ≥ 0.4762
The domain of the demand curve is [0.4762, ∞).
b. To find the market equilibrium, we set the quantity supplied equal to the quantity demanded. So we equate the two equations:
380 - 80p = 84p - 40
Simplifying, we get:
160p = 420
p = 2.625
Substituting this value back into either equation, we find:
x = 380 - 80(2.625)
x ≈ 218.75
Therefore, the market equilibrium is x = 218.75 and p = 2.625.
c. The revenue function (R(x)) represents the total revenue generated from selling a particular quantity (x) at a certain price (p). To find the revenue function, we multiply the quantity (x) by the unit price (p):
R(x) = x * p
Substituting the equilibrium values we found, the revenue function becomes:
R(x) = 218.75 * 2.625
d. To determine the revenue at market equilibrium, we evaluate the revenue function at x = 218.75:
R(x) = 218.75 * 2.625
R(x) ≈ 573.24
Therefore, the revenue at market equilibrium is approximately 573.24.