Sure, let's work through the problem step by step to find the producer's surplus when the market price is set at [tex]$10 per unit.
### Step 1: Understand the Supply Function
The supplier's supply function is given by:
\[ p = \sqrt{9 + 2.6x} \]
where \( p \) is the unit price and \( x \) is the quantity in hundreds of units.
### Step 2: Find the Quantity Supplied at the Market Price
We need to determine the quantity \( x \) for which the unit price \( p \) is $[/tex]10.
Set [tex]\( p = 10 \)[/tex]:
[tex]\[ 10 = \sqrt{9 + 2.6x} \][/tex]
Square both sides to eliminate the square root:
[tex]\[ 100 = 9 + 2.6x \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ 100 - 9 = 2.6x \][/tex]
[tex]\[ 91 = 2.6x \][/tex]
[tex]\[ x = \frac{91}{2.6} \][/tex]
[tex]\[ x \approx 35 \][/tex]
So, the quantity supplied when the market price is [tex]$10 per unit is approximately 35 hundred units, or 3500 units.
### Step 3: Producer's Surplus Concept
Producer's surplus is the difference between what producers are paid and what they are willing to accept. Mathematically, it's calculated as the area between the market price line and the supply curve from 0 to the quantity supplied.
### Step 4: Calculate the Integral of the Supply Function
To find the producer's surplus, we need to integrate the supply function from 0 to the quantity supplied, which is \( \int_0^{35} \sqrt{9 + 2.6x} \, dx \).
Let's denote this integral as \( I \).
### Step 5: Compute the Producer’s Surplus
The integral \( I \) represents the total area under the supply curve from 0 to 35. The producer's surplus is given by:
\[ \text{Producer's Surplus} = (\text{Market Price} \times \text{Quantity Supplied}) - I \]
Substituting the given values:
\[ \text{Market Price} = 10 \]
\[ \text{Quantity Supplied} \approx 35 \]
\[ I \approx 249.48717948715637 \]
Now, calculate the producer's surplus:
\[ \text{Producer's Surplus} = (10 \times 35) - 249.48717948715637 \]
\[ \text{Producer's Surplus} = 350 - 249.48717948715637 \]
\[ \text{Producer's Surplus} \approx 100.51 \]
### Final Answer
Thus, the producer's surplus when the market price is set at $[/tex]10 per unit is approximately $100.51.
[tex]\[ \boxed{100.51} \][/tex]
