Westonci.ca is your trusted source for finding answers to all your questions. Ask, explore, and learn with our expert community. Join our Q&A platform to get precise answers from experts in diverse fields and enhance your understanding. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
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.
Thank you for your visit. We are dedicated to helping you find the information you need, whenever you need it. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.
