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(49 + 4.4x) dollars. Determine the producers' surplus if the market price is set at $15/unit. (Round your answer to two decimal places.)
To determine the producer's surplus, we need to calculate the difference between the price at which the supplier is willing to sell the units and the market price.
Given:
Unit price from the supplier: p = sqrt(49 + 4.4x) dollars
Market price: $15/unit
To find the producers' surplus, we need to find the quantity of hair dryers that will be sold at the market price.
Setting the unit price equal to the market price:
sqrt(49 + 4.4x) = 15
We can now solve this equation to find the value of x.
1. Square both sides of the equation:
49 + 4.4x = 15^2
49 + 4.4x = 225
2. Subtract 49 from both sides:
4.4x = 225 - 49
4.4x = 176
3. Divide both sides by 4.4:
x = 176 / 4.4
x = 40
So, x = 40 hundred units of hair dryers will be sold at the market price.
To calculate the producer's surplus, we need to find the total revenue the supplier would get by selling x hundred units at the market price and subtract the cost of production.
The total revenue for x hundred units is given by:
Revenue = Price * Quantity
Revenue = $15 * 40
Revenue = $600
To find the cost of production, plug the value of x into the unit price equation:
p = sqrt(49 + 4.4x)
p = sqrt(49 + 4.4 * 40)
p = sqrt(49 + 176)
p = sqrt(225)
p = 15
The cost of production per unit is $15.
The cost of production for x hundred units is given by:
Cost of production = Cost per unit * Quantity
Cost of production = $15 * 40
Cost of production = $600
Now, we can finally calculate the producer's surplus by subtracting the cost of production from the total revenue:
Producer's Surplus = Revenue - Cost of production
Producer's Surplus = $600 - $600
Producer's Surplus = $0
Therefore, the producer's surplus is $0.