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].
