If the supply curve is \( p=\sqrt{9+x} \) and the quantity sold in market is 7 units, then the producer's surplus will be
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( \displaystyle\frac{10}{3} \)
Producer's surplus is the revenue at the market price minus the area under the supply curve:
\( \displaystyle PS=p_0x_0-\int_{0}^{x_0}S(x)\,dx \)
At \( x_0=7 \), \( p_0=\sqrt{9+7}=4 \), so \( p_0x_0=28 \).
\( \displaystyle \int_{0}^{7}\sqrt{9+x}\,dx=\left[\frac{2}{3}(9+x)^{3/2}\right]_{0}^{7} \)
\( \displaystyle =\frac{2}{3}\left(16^{3/2}-9^{3/2}\right) \)
\( \displaystyle =\frac{2}{3}(64-27) \)
\( \displaystyle =\frac{74}{3} \)
\( \displaystyle PS=28-\frac{74}{3} \)
\( \displaystyle =\frac{84-74}{3}=\frac{10}{3} \)