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Sagot :
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).
[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).
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