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.
466p+90x−2390=0
and
484p−22x−978=0
Determine the revenue function. Revenue function R(x)=?
I got (-90x/466) + (2390x/466), but apparently that is wrong...
I'd say revenue = price * demand
If the 2nd equation is the demand function, then we have
p = (22x+978)/484
revenue would thus be
R(x) = p*x = x(22x+978)/484
Not sure why the supply curve is given. Doesn't matter how many widgets you can dump on the market if no one wants them.
Am I off base here?
Actually I think you used the wrong function. The 1st one is the demand, not the second.
To determine the revenue function from the given pair of equations representing the supply and demand curves, we need to understand that revenue is calculated by multiplying the unit price (p) by the quantity (x) sold.
First, let's solve both equations simultaneously to find the equilibrium price and quantity, which represents the price and quantity at which supply equals demand. We can solve these equations by using the method of substitution or elimination.
Let's start with the equations:
466p + 90x - 2390 = 0 (Supply equation)
484p - 22x - 978 = 0 (Demand equation)
To solve them, we can rearrange the supply equation to solve for p:
466p = 2390 - 90x
p = (2390 - 90x) / 466
Now, substitute this value of p into the demand equation:
484[(2390 - 90x) / 466] - 22x - 978 = 0
Now, simplify and solve for x:
(484 * (2390 - 90x)) / 466 - 22x - 978 = 0
(484 * (2390 - 90x)) / 466 = 22x + 978
(484 * (2390 - 90x)) = (466 * (22x + 978))
1160960 - 43760x = 10252x + 454548
-54012x = -705412
x = 705412 / 54012
x ≈ 13.05
Now that we have the equilibrium quantity (x), substitute it back into the supply equation to find the equilibrium price (p):
p = (2390 - 90 * 13.05) / 466
p ≈ 3.71
Therefore, the equilibrium price (p) is approximately 3.71, and the equilibrium quantity (x) is approximately 13.05.
To find the revenue function, we can substitute these equilibrium values into either the supply or demand equation since they both represent the equilibrium point. Let's use the demand equation:
484p - 22x - 978 = 0
484 * 3.71 - 22 * 13.05 - 978 = 0
1800.44 - 287.1 - 978 = 0
-484.66 - 978 = 0
-1462.66 = 0
Therefore, the revenue function is R(x) = -1462.66.
Please note that the answer might vary slightly depending on rounding off decimals during calculations.