To find the equilibrium point for Max's T-shirt business, we need to determine the quantity of T-shirts sold (Q) and the price at which they are sold (P) at the point where the demand and supply functions intersect.
The demand function is given by:
[tex]\[ P = -Q + 34 \][/tex]
The supply function is given by:
[tex]\[ P = Q - 10 \][/tex]
At equilibrium, the quantity demanded equals the quantity supplied. Therefore, we set the two equations equal to each other:
[tex]\[ -Q + 34 = Q - 10 \][/tex]
Next, we solve for [tex]\(Q\)[/tex]:
1. Add [tex]\(Q\)[/tex] to both sides:
[tex]\[ 34 = 2Q - 10 \][/tex]
2. Add 10 to both sides:
[tex]\[ 44 = 2Q \][/tex]
3. Divide by 2:
[tex]\[ Q = 22 \][/tex]
Now that we have the equilibrium quantity, we substitute [tex]\(Q = 22\)[/tex] back into either the demand or supply function to find the equilibrium price [tex]\(P\)[/tex]. Using the supply function:
[tex]\[ P = Q - 10 \][/tex]
[tex]\[ P = 22 - 10 \][/tex]
[tex]\[ P = 12 \][/tex]
Therefore, the equilibrium point is [tex]\((22, 12)\)[/tex].
The correct answer is:
D. [tex]\((22, 12)\)[/tex]
