Suppose the supply function for a certain item is given by [tex]\( S(q) = (q+7)^2 \)[/tex] and the demand function is given by [tex]\( D(q) = \frac{729}{q+7} \)[/tex]. Complete parts (a) through (d) below.

a. Graph the supply and demand curves. Choose the correct graph below.
A.
B.
C.
D.

b. Find the point at which supply and demand are in equilibrium.

The equilibrium point is [tex]\( (2, 81) \)[/tex].
(Type an ordered pair. Simplify your answer.)

c. Find the consumers' surplus.

The consumers' surplus is [tex]\( \$21.21 \)[/tex].
(Round to two decimal places as needed.)

d. Find the producers' surplus.



Answer :

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.