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Sagot :
Sure, let's work through each part of this problem step by step.
### Part a: Graph the supply and demand curves
To graph the supply and demand curves, we need to understand the behavior of each function.
1. Supply Function: [tex]\( S(q) = (q + 7)^2 \)[/tex]
- This is a parabola opening upwards, shifted 7 units to the left.
2. Demand Function: [tex]\( D(q) = \frac{729}{q + 7} \)[/tex]
- This is a hyperbola that decreases as [tex]\( q \)[/tex] increases, shifted 7 units to the left.
With these characteristics, you should be able to choose the correct graph among the options provided.
### Part b: Find the point at which supply and demand are in equilibrium
The equilibrium point is found where the supply equals the demand, i.e., [tex]\( S(q) = D(q) \)[/tex].
To find this equilibrium point:
[tex]\[ (q + 7)^2 = \frac{729}{q + 7} \][/tex]
Solving this equation for [tex]\( q \)[/tex]:
[tex]\[ (q + 7)^3 = 729 \][/tex]
[tex]\[ q + 7 = \sqrt[3]{729} \][/tex]
[tex]\[ q + 7 = 9 \][/tex]
[tex]\[ q = 2 \][/tex]
To find the equilibrium price [tex]\( p \)[/tex], substitute [tex]\( q = 2 \)[/tex] back into either the supply or demand function:
[tex]\[ p = (2 + 7)^2 = 81 \][/tex]
Therefore, the equilibrium point is [tex]\( (2, 81) \)[/tex].
### Part c: Find the consumers' surplus
Consumers' surplus is the area between the demand curve and the price line from [tex]\( q = 0 \)[/tex] to the equilibrium quantity [tex]\( q = 2 \)[/tex]:
[tex]\[ \text{Consumers' Surplus} = \int_0^2 \left[ D(q) - p_e \right] \, dq \][/tex]
Using the equilibrium price [tex]\( p_e = 81 \)[/tex]:
[tex]\[ \text{Consumers' Surplus} = \int_0^2 \left( \frac{729}{q + 7} - 81 \right) \, dq \][/tex]
After evaluating this integral:
[tex]\[ \text{Consumers' Surplus} \approx 21.21 \][/tex]
### Part d: Find the producers' surplus
Producers' surplus is the area between the supply curve and the price line from [tex]\( q = 0 \)[/tex] to the equilibrium quantity [tex]\( q = 2 \)[/tex]:
[tex]\[ \text{Producers' Surplus} = \int_0^2 \left[ p_e - S(q) \right] \, dq \][/tex]
Using the equilibrium price [tex]\( p_e = 81 \)[/tex]:
[tex]\[ \text{Producers' Surplus} = \int_0^2 \left( 81 - (q + 7)^2 \right) \, dq \][/tex]
After evaluating this integral:
[tex]\[ \text{Producers' Surplus} \approx 33.33 \][/tex]
### Summary:
1. Equilibrium Point: [tex]\( (2, 81) \)[/tex]
2. Consumers' Surplus: \[tex]$21.21 (rounded to two decimal places) 3. Producers' Surplus: \$[/tex]33.33 (rounded to two decimal places)
These results describe the economic equilibrium of the given supply and demand functions.
### Part a: Graph the supply and demand curves
To graph the supply and demand curves, we need to understand the behavior of each function.
1. Supply Function: [tex]\( S(q) = (q + 7)^2 \)[/tex]
- This is a parabola opening upwards, shifted 7 units to the left.
2. Demand Function: [tex]\( D(q) = \frac{729}{q + 7} \)[/tex]
- This is a hyperbola that decreases as [tex]\( q \)[/tex] increases, shifted 7 units to the left.
With these characteristics, you should be able to choose the correct graph among the options provided.
### Part b: Find the point at which supply and demand are in equilibrium
The equilibrium point is found where the supply equals the demand, i.e., [tex]\( S(q) = D(q) \)[/tex].
To find this equilibrium point:
[tex]\[ (q + 7)^2 = \frac{729}{q + 7} \][/tex]
Solving this equation for [tex]\( q \)[/tex]:
[tex]\[ (q + 7)^3 = 729 \][/tex]
[tex]\[ q + 7 = \sqrt[3]{729} \][/tex]
[tex]\[ q + 7 = 9 \][/tex]
[tex]\[ q = 2 \][/tex]
To find the equilibrium price [tex]\( p \)[/tex], substitute [tex]\( q = 2 \)[/tex] back into either the supply or demand function:
[tex]\[ p = (2 + 7)^2 = 81 \][/tex]
Therefore, the equilibrium point is [tex]\( (2, 81) \)[/tex].
### Part c: Find the consumers' surplus
Consumers' surplus is the area between the demand curve and the price line from [tex]\( q = 0 \)[/tex] to the equilibrium quantity [tex]\( q = 2 \)[/tex]:
[tex]\[ \text{Consumers' Surplus} = \int_0^2 \left[ D(q) - p_e \right] \, dq \][/tex]
Using the equilibrium price [tex]\( p_e = 81 \)[/tex]:
[tex]\[ \text{Consumers' Surplus} = \int_0^2 \left( \frac{729}{q + 7} - 81 \right) \, dq \][/tex]
After evaluating this integral:
[tex]\[ \text{Consumers' Surplus} \approx 21.21 \][/tex]
### Part d: Find the producers' surplus
Producers' surplus is the area between the supply curve and the price line from [tex]\( q = 0 \)[/tex] to the equilibrium quantity [tex]\( q = 2 \)[/tex]:
[tex]\[ \text{Producers' Surplus} = \int_0^2 \left[ p_e - S(q) \right] \, dq \][/tex]
Using the equilibrium price [tex]\( p_e = 81 \)[/tex]:
[tex]\[ \text{Producers' Surplus} = \int_0^2 \left( 81 - (q + 7)^2 \right) \, dq \][/tex]
After evaluating this integral:
[tex]\[ \text{Producers' Surplus} \approx 33.33 \][/tex]
### Summary:
1. Equilibrium Point: [tex]\( (2, 81) \)[/tex]
2. Consumers' Surplus: \[tex]$21.21 (rounded to two decimal places) 3. Producers' Surplus: \$[/tex]33.33 (rounded to two decimal places)
These results describe the economic equilibrium of the given supply and demand functions.
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