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Sagot :
To determine the consumers' surplus and producers' surplus, let's go through the steps systematically:
1. Set the Demand Price equal to the Supply Price to find the Equilibrium Quantity:
The demand price equation is [tex]\( \rho = 140 - x^2 \)[/tex].
The supply price equation is [tex]\( \rho = 44 + \frac{1}{2} x^2 \)[/tex].
Set these two equations equal to each other to find the equilibrium quantity [tex]\( x \)[/tex].
[tex]\[ 140 - x^2 = 44 + \frac{1}{2} x^2 \][/tex]
Combine like terms:
[tex]\[ 140 - 44 = x^2 + \frac{1}{2} x^2 \][/tex]
[tex]\[ 96 = \frac{3}{2} x^2 \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ x^2 = \frac{96 \times 2}{3} = 64 \][/tex]
[tex]\[ x = \sqrt{64} = 8 \][/tex]
Therefore, the equilibrium quantity [tex]\( x = 8 \)[/tex] (in thousands of units).
2. Find the Equilibrium Price:
Substitute [tex]\( x = 8 \)[/tex] back into the demand equation to find the equilibrium price [tex]\( \rho \)[/tex].
[tex]\[ \rho = 140 - (8)^2 = 140 - 64 = 76 \][/tex]
Hence, the equilibrium price [tex]\( \rho = 76 \)[/tex] dollars.
3. Calculate Consumer Surplus:
Consumer surplus is the area between the demand curve and the equilibrium price up to the equilibrium quantity:
[tex]\[ \text{Consumer Surplus} = \int_0^8 (140 - x^2 - 76) \, dx \][/tex]
Simplify the integrand:
[tex]\[ 140 - x^2 - 76 = 64 - x^2 \][/tex]
Now, integrate:
[tex]\[ \text{Consumer Surplus} = \int_0^8 (64 - x^2) \, dx \][/tex]
Compute the integral:
[tex]\[ \int (64 - x^2) \, dx = 64x - \frac{x^3}{3} \][/tex]
Evaluate from 0 to 8:
[tex]\[ \left[ 64x - \frac{x^3}{3} \right]_0^8 = \left( 64 \cdot 8 - \frac{8^3}{3} \right) - \left( 64 \cdot 0 - \frac{0^3}{3} \right) \][/tex]
[tex]\[ = 512 - \frac{512}{3} \][/tex]
[tex]\[ = \frac{1536}{3} - \frac{512}{3} = \frac{1024}{3} \approx 341.33 \][/tex]
Rounded to the nearest dollar, the consumer surplus is approximately [tex]\( \$341 \)[/tex].
4. Calculate Producer Surplus:
Producer surplus is the area between the supply curve and the equilibrium price up to the equilibrium quantity:
[tex]\[ \text{Producer Surplus} = \int_0^8 (76 - (44 + \frac{1}{2} x^2)) \, dx \][/tex]
Simplify the integrand:
[tex]\[ 76 - 44 - \frac{1}{2} x^2 = 32 - \frac{1}{2} x^2 \][/tex]
Now, integrate:
[tex]\[ \text{Producer Surplus} = \int_0^8 (32 - \frac{1}{2} x^2) \, dx \][/tex]
Compute the integral:
[tex]\[ \int (32 - \frac{1}{2} x^2) \, dx = 32x - \frac{1}{6} x^3 \][/tex]
Evaluate from 0 to 8:
[tex]\[ \left[ 32x - \frac{1}{6} x^3 \right]_0^8 = \left( 32 \cdot 8 - \frac{8^3}{6} \right) - \left( 32 \cdot 0 - \frac{0^3}{6} \right) \][/tex]
[tex]\[ = 256 - \frac{512}{6} \][/tex]
[tex]\[ = 256 - \frac{256}{3} = \frac{768}{3} - \frac{256}{3} = \frac{512}{3} \approx 170.67 \][/tex]
Rounded to the nearest dollar, the producer surplus is approximately [tex]\( \$171 \)[/tex].
Therefore, the consumers' surplus is \[tex]$341, and the producers' surplus is \$[/tex]171.
1. Set the Demand Price equal to the Supply Price to find the Equilibrium Quantity:
The demand price equation is [tex]\( \rho = 140 - x^2 \)[/tex].
The supply price equation is [tex]\( \rho = 44 + \frac{1}{2} x^2 \)[/tex].
Set these two equations equal to each other to find the equilibrium quantity [tex]\( x \)[/tex].
[tex]\[ 140 - x^2 = 44 + \frac{1}{2} x^2 \][/tex]
Combine like terms:
[tex]\[ 140 - 44 = x^2 + \frac{1}{2} x^2 \][/tex]
[tex]\[ 96 = \frac{3}{2} x^2 \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ x^2 = \frac{96 \times 2}{3} = 64 \][/tex]
[tex]\[ x = \sqrt{64} = 8 \][/tex]
Therefore, the equilibrium quantity [tex]\( x = 8 \)[/tex] (in thousands of units).
2. Find the Equilibrium Price:
Substitute [tex]\( x = 8 \)[/tex] back into the demand equation to find the equilibrium price [tex]\( \rho \)[/tex].
[tex]\[ \rho = 140 - (8)^2 = 140 - 64 = 76 \][/tex]
Hence, the equilibrium price [tex]\( \rho = 76 \)[/tex] dollars.
3. Calculate Consumer Surplus:
Consumer surplus is the area between the demand curve and the equilibrium price up to the equilibrium quantity:
[tex]\[ \text{Consumer Surplus} = \int_0^8 (140 - x^2 - 76) \, dx \][/tex]
Simplify the integrand:
[tex]\[ 140 - x^2 - 76 = 64 - x^2 \][/tex]
Now, integrate:
[tex]\[ \text{Consumer Surplus} = \int_0^8 (64 - x^2) \, dx \][/tex]
Compute the integral:
[tex]\[ \int (64 - x^2) \, dx = 64x - \frac{x^3}{3} \][/tex]
Evaluate from 0 to 8:
[tex]\[ \left[ 64x - \frac{x^3}{3} \right]_0^8 = \left( 64 \cdot 8 - \frac{8^3}{3} \right) - \left( 64 \cdot 0 - \frac{0^3}{3} \right) \][/tex]
[tex]\[ = 512 - \frac{512}{3} \][/tex]
[tex]\[ = \frac{1536}{3} - \frac{512}{3} = \frac{1024}{3} \approx 341.33 \][/tex]
Rounded to the nearest dollar, the consumer surplus is approximately [tex]\( \$341 \)[/tex].
4. Calculate Producer Surplus:
Producer surplus is the area between the supply curve and the equilibrium price up to the equilibrium quantity:
[tex]\[ \text{Producer Surplus} = \int_0^8 (76 - (44 + \frac{1}{2} x^2)) \, dx \][/tex]
Simplify the integrand:
[tex]\[ 76 - 44 - \frac{1}{2} x^2 = 32 - \frac{1}{2} x^2 \][/tex]
Now, integrate:
[tex]\[ \text{Producer Surplus} = \int_0^8 (32 - \frac{1}{2} x^2) \, dx \][/tex]
Compute the integral:
[tex]\[ \int (32 - \frac{1}{2} x^2) \, dx = 32x - \frac{1}{6} x^3 \][/tex]
Evaluate from 0 to 8:
[tex]\[ \left[ 32x - \frac{1}{6} x^3 \right]_0^8 = \left( 32 \cdot 8 - \frac{8^3}{6} \right) - \left( 32 \cdot 0 - \frac{0^3}{6} \right) \][/tex]
[tex]\[ = 256 - \frac{512}{6} \][/tex]
[tex]\[ = 256 - \frac{256}{3} = \frac{768}{3} - \frac{256}{3} = \frac{512}{3} \approx 170.67 \][/tex]
Rounded to the nearest dollar, the producer surplus is approximately [tex]\( \$171 \)[/tex].
Therefore, the consumers' surplus is \[tex]$341, and the producers' surplus is \$[/tex]171.
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