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Max's T-shirt business uses the demand function [tex]P=-Q+34[/tex] and the supply function [tex]P=Q-10[/tex]. According to these functions, what will the equilibrium point be for Max's T-shirt business (i.e., the number of T-shirts sold and the price at which they're sold)?

A. [tex](22,12)[/tex]
B. [tex](10,34)[/tex]
C. [tex](34,10)[/tex]
D. [tex](12,22)[/tex]


Sagot :

To find the equilibrium point for Max's T-shirt business, we need to determine the point where the demand and supply functions intersect. This means finding the values of [tex]\(Q\)[/tex] (quantity) and [tex]\(P\)[/tex] (price) that satisfy both equations simultaneously.

The demand function is given by:
[tex]\[ P = -Q + 34 \][/tex]

And the supply function is given by:
[tex]\[ P = Q - 10 \][/tex]

To find the equilibrium point, we equate the two expressions for [tex]\(P\)[/tex] from the demand and supply functions and solve for [tex]\(Q\)[/tex]:

[tex]\[ -Q + 34 = Q - 10 \][/tex]

Now, let's solve this equation step-by-step:

1. Add [tex]\(Q\)[/tex] to both sides to move the [tex]\(Q\)[/tex] term to one side:
[tex]\[ 34 = 2Q - 10 \][/tex]

2. Add 10 to both sides to isolate the [tex]\(2Q\)[/tex] term:
[tex]\[ 34 + 10 = 2Q \][/tex]
[tex]\[ 44 = 2Q \][/tex]

3. Divide both sides by 2 to solve for [tex]\(Q\)[/tex]:
[tex]\[ Q = \frac{44}{2} \][/tex]
[tex]\[ Q = 22 \][/tex]

Now that we have the quantity [tex]\(Q = 22\)[/tex], we need to find the corresponding price [tex]\(P\)[/tex]. We can use either the demand or supply function to do this. Let's use the supply function:
[tex]\[ P = Q - 10 \][/tex]

Substitute [tex]\(Q = 22\)[/tex] into the supply function:
[tex]\[ P = 22 - 10 \][/tex]
[tex]\[ P = 12 \][/tex]

Therefore, the equilibrium point for Max's T-shirt business is [tex]\((22, 12)\)[/tex].

The correct answer is:
A. [tex]\((22, 12)\)[/tex]