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.)
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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 find the difference between what the supplier receives from selling the hair dryers and what they would have been willing to accept.
Given that the market price is set at $15/unit, we need to calculate the quantity demanded (x) at this price.
Let's start by setting the unit price equation equal to $15 and solving for x:
sqrt(49 + 4.4x) = 15
Squaring both sides:
49 + 4.4x = 15^2
49 + 4.4x = 225
Subtracting 49 from both sides:
4.4x = 225 - 49
4.4x = 176
Dividing both sides by 4.4:
x = 176 / 4.4
x = 40
Therefore, the quantity demanded at a price of $15/unit is 40 hundred units.
To find the producer's surplus, we need to calculate the revenue the supplier receives at this quantity demanded.
Revenue = Price * Quantity Demanded
Revenue = $15/unit * 40 hundred units
Converting hundred units to regular units by multiplying by 100:
Revenue = $15/unit * 40 * 100 units
Revenue = $60,000
Next, we need to find the total cost the supplier incurs to produce x hundred units of hair dryers. The cost for each unit is given by the unit price equation.
Unit Cost = sqrt(49 + 4.4x) dollars
Substituting the quantity demanded, x = 40, into the unit price equation:
Unit Cost = sqrt(49 + 4.4 * 40) dollars
Unit Cost = sqrt(49 + 176) dollars
Unit Cost = sqrt(225) dollars
Unit Cost = $15
Therefore, the unit cost is $15/unit.
Total Cost = Unit Cost * Quantity Demanded
Total Cost = $15/unit * 40 hundred units * 100 units/hundred units
Converting hundred units to regular units:
Total Cost = $15/unit * 4,000 units
Total Cost = $60,000
Finally, we can calculate the producer's surplus as the difference between revenue and total cost:
Producer's Surplus = Revenue - Total Cost
Producer's Surplus = $60,000 - $60,000
Producer's Surplus = $0
Therefore, the producer's surplus is $0.
To determine the producer's surplus, we need to calculate the difference between the total revenue received by the supplier and the cost of producing the hair dryers.
The total revenue is given by the market price per unit multiplied by the quantity sold. In this case, the market price is $15/unit.
To find the quantity sold, we need to solve for x in the given equation:
p = sqrt(49 + 4.4x) = 15
Taking the square of both sides, we have:
49 + 4.4x = 225
Now, solving for x:
4.4x = 225 - 49
4.4x = 176
x = 40
Therefore, the supplier will make 40 hundred units (or 4000 units) of hair dryers available in the market.
Now, let's calculate the producer's surplus.
The cost of producing the hair dryers is the total cost of manufacturing x hundred units. However, the cost function is not given in the question. Without additional information about the cost, we cannot determine the exact producer's surplus.