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## Sagot :

1.

**Understand the given values:**

- Fixed cost to keep the online shop active: [tex]\( \$ 500 \)[/tex] per month.

- Marginal cost (cost per additional shirt produced/purchased): [tex]\( \$ 5 \)[/tex] per shirt.

- Marginal benefit (revenue per additional shirt sold): [tex]\( \$ 10 \)[/tex] per shirt.

2.

**Identify the goal:**

- We want to find the quantity of shirts sold where the total cost is equal to the total benefit.

3.

**Define the relationships:**

- Total cost is given by [tex]\( \text{Total Cost} = \text{Fixed Cost} + \text{Marginal Cost per Shirt} \times \text{Quantity Sold} \)[/tex].

- Total benefit is given by [tex]\( \text{Total Benefit} = \text{Marginal Benefit per Shirt} \times \text{Quantity Sold} \)[/tex].

4.

**Set up the equations:**

- We have a fixed cost of [tex]\( \$ 500 \)[/tex].

- We have a marginal cost of [tex]\( \$ 5 \)[/tex] per shirt.

- We have a marginal benefit of [tex]\( \$ 10 \)[/tex] per shirt.

Therefore:

[tex]\[ \text{Total Cost} = 500 + 5 \times Q \][/tex]

[tex]\[ \text{Total Benefit} = 10 \times Q \][/tex]

5.

**Set the total cost equal to the total benefit:**

[tex]\[ 500 + 5Q = 10Q \][/tex]

6.

**Solve for [tex]\( Q \)[/tex] (Quantity of shirts):**

- Subtract [tex]\( 5Q \)[/tex] from both sides:

[tex]\[ 500 = 5Q \][/tex]

- Divide both sides by 5:

[tex]\[ Q = 100 \][/tex]

7.

**Verify the solution:**

- When [tex]\( Q = 100 \)[/tex]:

[tex]\[ \text{Total Cost} = 500 + 5 \times 100 = 1000 \][/tex]

[tex]\[ \text{Total Benefit} = 10 \times 100 = 1000 \][/tex]

So, the retailer needs to sell

**100 shirts**for the total costs to be equal to the total benefits. This means at [tex]\( Q = 100 \)[/tex] shirts, both the total cost and total benefit are [tex]\( \$ 1000 \)[/tex].