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To determine which equation represents the profit function of the newsstand, let's break down the problem step-by-step.
1. Fixed Costs:
The newsstand spends a fixed amount of \[tex]$600 a month on rent and electricity. This amount does not change regardless of how many magazines are sold. Therefore, the fixed cost is \$[/tex]600.
2. Variable Costs:
The cost of each magazine is \[tex]$2. If the newsstand sells \( n \) magazines, the total variable cost will be \$[/tex]2 multiplied by the number of magazines sold, i.e., \[tex]$2n. 3. Revenue: The newsstand charges \$[/tex]5 for each magazine sold. Therefore, the revenue generated from selling [tex]\( n \)[/tex] magazines is \[tex]$5 multiplied by \( n \), i.e., \$[/tex]5n.
4. Profit Calculation:
Profit is calculated by subtracting the total costs (both fixed and variable) from the total revenue. Therefore, we need to derive the profit function using the following formula:
[tex]\[ \text{Profit} = \text{Revenue} - \text{Total Costs} \][/tex]
5. Calculating the Profit:
- Total revenue from selling [tex]\( n \)[/tex] magazines: \[tex]$5n - Total variable cost from purchasing \( n \) magazines: \$[/tex]2n
- Total fixed cost: \[tex]$600 - Hence, the total costs (fixed + variable) are: \$[/tex]600 + \$2n
Therefore, the profit function [tex]\( p \)[/tex] can be written as:
[tex]\[ p = 5n - (600 + 2n) \][/tex]
6. Simplifying the Profit Function:
Simplify the expression:
[tex]\[ p = 5n - 600 - 2n \][/tex]
Combine the like terms:
[tex]\[ p = (5n - 2n) - 600 \][/tex]
[tex]\[ p = 3n - 600 \][/tex]
So, the equation that represents the profit function of the newsstand is:
[tex]\[ p = 3n - 600 \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{D} p = 3n - 600 \][/tex]
1. Fixed Costs:
The newsstand spends a fixed amount of \[tex]$600 a month on rent and electricity. This amount does not change regardless of how many magazines are sold. Therefore, the fixed cost is \$[/tex]600.
2. Variable Costs:
The cost of each magazine is \[tex]$2. If the newsstand sells \( n \) magazines, the total variable cost will be \$[/tex]2 multiplied by the number of magazines sold, i.e., \[tex]$2n. 3. Revenue: The newsstand charges \$[/tex]5 for each magazine sold. Therefore, the revenue generated from selling [tex]\( n \)[/tex] magazines is \[tex]$5 multiplied by \( n \), i.e., \$[/tex]5n.
4. Profit Calculation:
Profit is calculated by subtracting the total costs (both fixed and variable) from the total revenue. Therefore, we need to derive the profit function using the following formula:
[tex]\[ \text{Profit} = \text{Revenue} - \text{Total Costs} \][/tex]
5. Calculating the Profit:
- Total revenue from selling [tex]\( n \)[/tex] magazines: \[tex]$5n - Total variable cost from purchasing \( n \) magazines: \$[/tex]2n
- Total fixed cost: \[tex]$600 - Hence, the total costs (fixed + variable) are: \$[/tex]600 + \$2n
Therefore, the profit function [tex]\( p \)[/tex] can be written as:
[tex]\[ p = 5n - (600 + 2n) \][/tex]
6. Simplifying the Profit Function:
Simplify the expression:
[tex]\[ p = 5n - 600 - 2n \][/tex]
Combine the like terms:
[tex]\[ p = (5n - 2n) - 600 \][/tex]
[tex]\[ p = 3n - 600 \][/tex]
So, the equation that represents the profit function of the newsstand is:
[tex]\[ p = 3n - 600 \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{D} p = 3n - 600 \][/tex]
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