Assume that the demand curve [tex]\( D(p) \)[/tex] given below is the market demand for widgets:
[tex]\[ Q = D(p) = 1117 - 12p, \quad p \ \textgreater \ 0 \][/tex]

Let the market supply of widgets be given by:
[tex]\[ Q = S(p) = -5 + 10p, \quad p \ \textgreater \ 0 \][/tex]

where [tex]\( p \)[/tex] is the price and [tex]\( Q \)[/tex] is the quantity. The functions [tex]\( D(p) \)[/tex] and [tex]\( S(p) \)[/tex] give the number of widgets demanded and supplied at a given price.

1. What is the equilibrium price? (Please round your answer to the nearest hundredth.)
2. What is the equilibrium quantity? (Please round your answer to the nearest integer.)
3. What is the total revenue at equilibrium? (Please round your answer to the nearest integer.)



Answer :

Let's solve the given problem step-by-step:

1. Determine the equilibrium price:

The equilibrium price is where the quantity demanded equals the quantity supplied. This occurs when [tex]\(D(p) = S(p)\)[/tex].

Given:
[tex]\[ D(p) = 1117 - 12p \][/tex]
[tex]\[ S(p) = -5 + 10p \][/tex]

To find the equilibrium price [tex]\( p \)[/tex], we set [tex]\( D(p) \)[/tex] equal to [tex]\( S(p) \)[/tex]:

[tex]\[ 1117 - 12p = -5 + 10p \][/tex]

Next, rearrange the equation to isolate [tex]\( p \)[/tex]:

[tex]\[ 1117 + 5 = 10p + 12p \][/tex]

Combine like terms:

[tex]\[ 1122 = 22p \][/tex]

Now solve for [tex]\( p \)[/tex]:

[tex]\[ p = \frac{1122}{22} \][/tex]

We find that the equilibrium price is:

[tex]\[ p = 51.0 \][/tex]

2. Determine the equilibrium quantity:

We can find the equilibrium quantity by substituting the equilibrium price back into either the demand equation [tex]\( D(p) \)[/tex] or the supply equation [tex]\( S(p) \)[/tex]. Let's use the demand equation:

[tex]\[ Q = D(51.0) = 1117 - 12 \times 51.0 \][/tex]

Calculate the equilibrium quantity:

[tex]\[ Q = 1117 - 612 \][/tex]
[tex]\[ Q = 505 \][/tex]

So, the equilibrium quantity is:

[tex]\[ Q = 505 \][/tex]

3. Determine the total revenue at equilibrium:

Total revenue is calculated by multiplying the equilibrium price by the equilibrium quantity:

[tex]\[ \text{Total Revenue} = \text{price} \times \text{quantity} \][/tex]
[tex]\[ \text{Total Revenue} = 51.0 \times 505 \][/tex]

Calculate the total revenue:

[tex]\[ \text{Total Revenue} = 25755 \][/tex]

Therefore, at equilibrium:

- The equilibrium price is [tex]\( \boxed{51.0} \)[/tex].
- The equilibrium quantity is [tex]\( \boxed{505} \)[/tex].
- The total revenue at equilibrium is [tex]\( \boxed{25755} \)[/tex].