A supplier of portable hair dryers will make x hundred units of hair dryers available in the market when the unit price is
p = sqrt(324 + 4x)
dollars. Determine the producers' surplus if the market price is set at $22/unit. (Round your answer to two decimal places.)
To determine the producer's surplus, we need to find the difference between the market price and the cost of producing the hair dryers.
Given that the market price is set at $22/unit, we can substitute this value into the equation for the unit price:
p = √(324 + 4x)
So, we have:
22 = √(324 + 4x)
To solve for x, we need to square both sides of the equation:
(22)^2 = ( √(324 + 4x) )^2
484 = 324 + 4x
Now, rearrange the equation to isolate x:
4x = 484 - 324
4x = 160
x = 160/4
x = 40
Therefore, when the market price is set at $22/unit, the supplier will make 40 hundred units of hair dryers available in the market.
To determine the producer's surplus, we need to calculate the difference between the cost of producing these units and the revenue generated at the market price.
The cost of producing x hundred units can be calculated using the unit price function:
p = √(324 + 4x)
Substituting x = 40 into the equation, we have:
p = √(324 + 4(40))
= √(324 + 160)
= √484
= 22
So, the cost of producing each unit is $22.
Now, we can calculate the revenue generated by multiplying the market price by the number of units:
Revenue = Market Price * Number of Units
= $22/unit * 40 hundred units
= $880
The producer's surplus is the difference between the revenue generated and the cost of production:
Producer's Surplus = Revenue - Cost of Production
= $880 - ($22/unit * 40 hundred units)
= $880 - $880
= $0
Therefore, the producer's surplus when the market price is set at $22/unit is $0.