Question
Is the total profit from selling t-shirts at a fixed price proportional to the number of t-shirts sold, given a fixed initial cost for supplies?
Answers
GPT-4o mini
Yes, the total profit from selling t-shirts at a fixed price is proportional to the number of t-shirts sold, assuming a fixed initial cost for supplies.
Let's break it down:
1. **Define Profit**: Profit is calculated as total revenue minus total costs.
2. **Total Revenue**: If you sell t-shirts at a fixed price (let's call it P) and sell N t-shirts, the total revenue (R) is given by:
\[
R = P \times N
\]
3. **Total Costs**: If you have a fixed initial cost for supplies (let's call it C), the total costs (T) remain constant regardless of the number of t-shirts sold:
\[
T = C
\]
4. **Total Profit**: The total profit (profit) can be expressed as:
\[
\text{Profit} = R - T = (P \times N) - C
\]
5. **Profit as a Function of N**: Rearranging the equation gives:
\[
\text{Profit} = P \times N - C
\]
In this equation, the term \(P \times N\) indicates that profit increases linearly with the number of t-shirts sold (N), while C is a fixed cost. The linear relationship implies that profit is indeed proportional to the number of t-shirts sold after covering the initial fixed cost.
However, it should be noted that while the total profit is proportional to the number of t-shirts sold, the profit per t-shirt sold does not change. The concept of proportionality here refers to the relationship between profit and the quantity sold over the range of sales considered, assuming the fixed price and fixed costs remain unchanged.
Let's break it down:
1. **Define Profit**: Profit is calculated as total revenue minus total costs.
2. **Total Revenue**: If you sell t-shirts at a fixed price (let's call it P) and sell N t-shirts, the total revenue (R) is given by:
\[
R = P \times N
\]
3. **Total Costs**: If you have a fixed initial cost for supplies (let's call it C), the total costs (T) remain constant regardless of the number of t-shirts sold:
\[
T = C
\]
4. **Total Profit**: The total profit (profit) can be expressed as:
\[
\text{Profit} = R - T = (P \times N) - C
\]
5. **Profit as a Function of N**: Rearranging the equation gives:
\[
\text{Profit} = P \times N - C
\]
In this equation, the term \(P \times N\) indicates that profit increases linearly with the number of t-shirts sold (N), while C is a fixed cost. The linear relationship implies that profit is indeed proportional to the number of t-shirts sold after covering the initial fixed cost.
However, it should be noted that while the total profit is proportional to the number of t-shirts sold, the profit per t-shirt sold does not change. The concept of proportionality here refers to the relationship between profit and the quantity sold over the range of sales considered, assuming the fixed price and fixed costs remain unchanged.