To find the equilibrium price for the T-shirts at the concert, we need to determine the price at which the quantity supplied [tex]\( S \)[/tex] is equal to the quantity demanded [tex]\( D \)[/tex]. This involves solving for the price [tex]\( p \)[/tex] in the equations given by the supply and demand functions:
[tex]\[
S(p) = -250 + 45p
\][/tex]
[tex]\[
D(p) = 1000 - 80p
\][/tex]
The equilibrium price occurs where the quantity supplied equals the quantity demanded:
[tex]\[
-250 + 45p = 1000 - 80p
\][/tex]
To solve for [tex]\( p \)[/tex], first combine like terms and solve the equation step-by-step:
1. Add [tex]\( 80p \)[/tex] to both sides to isolate [tex]\( p \)[/tex]:
[tex]\[
-250 + 45p + 80p = 1000
\][/tex]
[tex]\[
-250 + 125p = 1000
\][/tex]
2. Add 250 to both sides to further isolate the term involving [tex]\( p \)[/tex]:
[tex]\[
125p = 1000 + 250
\][/tex]
[tex]\[
125p = 1250
\][/tex]
3. Finally, divide both sides by 125 to solve for [tex]\( p \)[/tex]:
[tex]\[
p = \frac{1250}{125}
\][/tex]
[tex]\[
p = 10
\][/tex]
Thus, the equilibrium price is [tex]\(\$10\)[/tex].
Therefore, the equilibrium price is:
[tex]\[
\$ \boxed{10}
\][/tex]
(rounded to the nearest dollar, as required).
