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(81 + 1.6x)
dollars. Determine the producers' surplus if the market price is set at $11/unit. (Round your answer to two decimal places.)
To determine the producers' surplus, we need to calculate the difference between the market price and the unit cost of production, and then multiply it by the number of units sold.
Given:
Unit price formula: p = sqrt(81 + 1.6x) dollars
Market price: $11/unit
To find x (the number of units sold):
11 = sqrt(81 + 1.6x)
Square both sides to eliminate the square root:
121 = 81 + 1.6x
Subtract 81 from both sides:
40 = 1.6x
Divide both sides by 1.6:
x = 25
Now that we know x = 25, we can substitute it into the unit price formula:
p = sqrt(81 + 1.6(25))
p = sqrt(81 + 40)
p = sqrt(121)
p = 11
The unit price calculated using the formula is the same as the market price, $11/unit.
To find the producer's surplus, we need to calculate the cost of production and subtract it from the revenue. In this case, since the market price and unit cost are the same, the producer's surplus is zero.
Therefore, the producer's surplus is $0.
To determine the producer's surplus, we need to find the difference between the total amount the supplier receives and the total amount the supplier would be willing to accept.
The total amount received by the supplier can be calculated by multiplying the market price per unit by the number of units sold. In this case, the market price is $11/unit.
To find the number of units sold, we need to equate the given market price with the unit price equation provided: p = sqrt(81 + 1.6x) dollars.
Setting p to $11, we have:
$11 = sqrt(81 + 1.6x)
Now we can solve for x:
Square both sides of the equation:
121 = 81 + 1.6x
Subtract 81 from both sides:
40 = 1.6x
Divide both sides by 1.6:
25 = x
So, x = 25.
Therefore, if the market price is set at $11/unit, the supplier will make 25 hundred units (or 2500 units) of hair dryers available in the market.
Now, we can calculate the producer's surplus:
The total amount the supplier receives is the market price per unit multiplied by the number of units sold:
Total amount received = $11/unit * 2500 units = $27,500
To find the total amount the supplier would be willing to accept, we need to substitute x = 25 into the unit price equation:
p = sqrt(81 + 1.6x) dollars
p = sqrt(81 + 1.6 * 25) dollars
p = sqrt(81 + 40) dollars
p = sqrt(121) dollars
p = $11
Therefore, the total amount the supplier would be willing to accept is also $11/unit.
The producer's surplus is the difference between the total amount received and the total amount the supplier would be willing to accept:
Producer's Surplus = Total amount received - Total amount willing to accept
= $27,500 - $27,500
= $0
Hence, the producer's surplus in this case is $0.
p=sqrt(81+1.6x)
=11.78
121 = 81 + 1.6x
121-81 = 1.6x
40 = 1.6x
25 = x
25(11) - int sqrt(81+ 1.6x)dx on [0. 25)
275 -250.833 = 24.17