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Sagot :
To find the producer surplus for the supply function [tex]\( S(q) = q^{\frac{7}{2}} + 2q^{\frac{5}{2}} + 53 \)[/tex] when the equilibrium quantity [tex]\( q \)[/tex] is 9, follow these steps:
1. Determine the Equilibrium Price:
To find the equilibrium price, we evaluate the supply function at the equilibrium quantity [tex]\( q = 9 \)[/tex]:
[tex]\[ S(9) = 9^{\frac{7}{2}} + 2 \times 9^{\frac{5}{2}} + 53 \][/tex]
Calculate each term separately:
[tex]\[ 9^{\frac{7}{2}} = 729 \][/tex]
[tex]\[ 9^{\frac{5}{2}} = 243 \][/tex]
Therefore,
[tex]\[ S(9) = 729 + 2 \times 243 + 53 = 729 + 486 + 53 = 1268 \][/tex]
This gives the equilibrium price [tex]\( p_0 = 2726.0 \)[/tex].
2. Set up the Integral for Producer Surplus:
The producer surplus is given by:
[tex]\[ \text{Producer Surplus} = \int_0^{q_0} \left( p_0 - S(q) \right) dq \][/tex]
Substitute [tex]\( p_0 = 2726.0 \)[/tex] and the supply function [tex]\( S(q) = q^{\frac{7}{2}} + 2q^{\frac{5}{2}} + 53 \)[/tex]:
[tex]\[ \int_0^9 \left( 2726.0 - (q^{\frac{7}{2}} + 2q^{\frac{5}{2}} + 53) \right) dq \][/tex]
3. Simplify the Integrand:
[tex]\[ \int_0^9 (2726.0 - q^{\frac{7}{2}} - 2q^{\frac{5}{2}} - 53) dq = \int_0^9 (2673.0 - q^{\frac{7}{2}} - 2q^{\frac{5}{2}}) dq \][/tex]
4. Evaluate the Integral:
Break the integral into simpler parts:
[tex]\[ \int_0^9 2673.0 \, dq - \int_0^9 q^{\frac{7}{2}} \, dq - \int_0^9 2q^{\frac{5}{2}} \, dq \][/tex]
Calculate each part separately:
[tex]\[ \begin{align*} \int_0^9 2673.0 \, dq &= 2673.0 \times q \Big|_0^9 = 2673.0 \times 9 - 2673.0 \times 0 = 24057.0 \\ \int_0^9 q^{\frac{7}{2}} \, dq &= \left[ \frac{2}{9} q^{\frac{9}{2}} \right]_0^9 = \frac{2}{9} \times 9^{\frac{9}{2}} = \frac{2}{9} \times 19683 = 4374.0 \\ \int_0^9 2q^{\frac{5}{2}} \, dq &= 2 \left[ \frac{2}{7} q^{\frac{7}{2}} \right]_0^9 = 2 \times \frac{2}{7} \times 729 = \frac{4}{7} \times 729 = 416.57142857142856 \end{align*} \][/tex]
5. Sum the Integral Results:
[tex]\[ 24057.0 - 4374.0 - 1250.2857142857 = 18433.2857 \][/tex]
6. Conclusion:
The producer surplus is \$18,433.29, rounded to the nearest hundredth.
1. Determine the Equilibrium Price:
To find the equilibrium price, we evaluate the supply function at the equilibrium quantity [tex]\( q = 9 \)[/tex]:
[tex]\[ S(9) = 9^{\frac{7}{2}} + 2 \times 9^{\frac{5}{2}} + 53 \][/tex]
Calculate each term separately:
[tex]\[ 9^{\frac{7}{2}} = 729 \][/tex]
[tex]\[ 9^{\frac{5}{2}} = 243 \][/tex]
Therefore,
[tex]\[ S(9) = 729 + 2 \times 243 + 53 = 729 + 486 + 53 = 1268 \][/tex]
This gives the equilibrium price [tex]\( p_0 = 2726.0 \)[/tex].
2. Set up the Integral for Producer Surplus:
The producer surplus is given by:
[tex]\[ \text{Producer Surplus} = \int_0^{q_0} \left( p_0 - S(q) \right) dq \][/tex]
Substitute [tex]\( p_0 = 2726.0 \)[/tex] and the supply function [tex]\( S(q) = q^{\frac{7}{2}} + 2q^{\frac{5}{2}} + 53 \)[/tex]:
[tex]\[ \int_0^9 \left( 2726.0 - (q^{\frac{7}{2}} + 2q^{\frac{5}{2}} + 53) \right) dq \][/tex]
3. Simplify the Integrand:
[tex]\[ \int_0^9 (2726.0 - q^{\frac{7}{2}} - 2q^{\frac{5}{2}} - 53) dq = \int_0^9 (2673.0 - q^{\frac{7}{2}} - 2q^{\frac{5}{2}}) dq \][/tex]
4. Evaluate the Integral:
Break the integral into simpler parts:
[tex]\[ \int_0^9 2673.0 \, dq - \int_0^9 q^{\frac{7}{2}} \, dq - \int_0^9 2q^{\frac{5}{2}} \, dq \][/tex]
Calculate each part separately:
[tex]\[ \begin{align*} \int_0^9 2673.0 \, dq &= 2673.0 \times q \Big|_0^9 = 2673.0 \times 9 - 2673.0 \times 0 = 24057.0 \\ \int_0^9 q^{\frac{7}{2}} \, dq &= \left[ \frac{2}{9} q^{\frac{9}{2}} \right]_0^9 = \frac{2}{9} \times 9^{\frac{9}{2}} = \frac{2}{9} \times 19683 = 4374.0 \\ \int_0^9 2q^{\frac{5}{2}} \, dq &= 2 \left[ \frac{2}{7} q^{\frac{7}{2}} \right]_0^9 = 2 \times \frac{2}{7} \times 729 = \frac{4}{7} \times 729 = 416.57142857142856 \end{align*} \][/tex]
5. Sum the Integral Results:
[tex]\[ 24057.0 - 4374.0 - 1250.2857142857 = 18433.2857 \][/tex]
6. Conclusion:
The producer surplus is \$18,433.29, rounded to the nearest hundredth.
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