To determine whether the total profit is proportional to the number of t-shirts you sell, let's define the variables involved.
- Let \( n \) be the number of t-shirts sold.
- Selling price per t-shirt: $12
- Cost for supplies: $40 (fixed cost)
Revenue from selling \( n \) t-shirts: This is calculated as the selling price multiplied by the number of t-shirts sold.
\[ \text{Revenue} = 12n \]
Total profit: This is calculated by subtracting the total costs from the revenue. Since the fixed cost for supplies is $40, the total costs when selling \( n \) t-shirts is still $40 (ignoring variable costs).
\[ \text{Total Profit} = \text{Revenue} - \text{Total Costs} \] \[ \text{Total Profit} = 12n - 40 \]
Now let's analyze the profit formula:
\[ \text{Total Profit} = 12n - 40 \]
This equation can be rearranged to:
\[ \text{Total Profit} = 12n - 40 = 12(n - \frac{40}{12}) \]
The expression \( 12(n - \frac{40}{12}) \) shows that the total profit increases with \( n \), but it also has a fixed offset of -40.
In relation to whether it is proportional to \( n \), we can make the following observation:
- Proportional relationship: A situation is proportional if it can be expressed in the form \( y = kx \), where \( k \) is a constant and \( x \) is the independent variable. Here, the profit equation is not simply in the form of \( k \cdot n \) because of the -40 term (the fixed costs).
Thus, since the profit does not directly scale linearly with \( n \) alone (due to the -40 fixed cost term), we conclude that the total profit is NOT proportional to the number of t-shirts sold. The profit only becomes positive once you've sold enough t-shirts to cover the initial cost. Specifically, you need to sell at least \( \frac{40}{12} \approx 3.33 \) t-shirts, or at least 4, to start making a profit.