Let's carefully go through each part of the problem, step-by-step.
1. Cost Equation:
Given:
- Fixed costs per week: \[tex]$100
- Variable cost per item: \$[/tex]2 per item
The total cost is the sum of the fixed costs and the variable costs, which can be written as:
[tex]\[ y = 100 + 2x \][/tex]
So, the cost equation in the slope-intercept form is:
[tex]\[ y = 100 + 2x \][/tex]
2. Revenue Equation:
Given:
- Selling price per item: \[tex]$15
The total revenue is the selling price multiplied by the number of items sold, which can be written as:
\[ y = 15x \]
So, the revenue equation in the slope-intercept form is:
\[ y = 15x \]
3. Total Costs When 10 Items Are Made:
The number of items produced and sold this week: \( x = 10 \)
Using the cost equation \( y = 100 + 2x \):
\[ y = 100 + 2(10) \]
\[ y = 100 + 20 \]
\[ y = 120 \]
Hence, the total costs to the retailer are:
\[ \$[/tex]120 \]
4. Total Revenue from Selling 10 Items:
Using the revenue equation [tex]\( y = 15x \)[/tex]:
[tex]\[ y = 15(10) \][/tex]
[tex]\[ y = 150 \][/tex]
Hence, the total revenue from selling 10 items is:
[tex]\[ \$150 \][/tex]
5. Profit Calculation:
Profit is the difference between the total revenue and the total costs. Therefore:
[tex]\[ \text{Profit} = \text{Total Revenue} - \text{Total Costs} \][/tex]
[tex]\[ \text{Profit} = 150 - 120 \][/tex]
[tex]\[ \text{Profit} = 30 \][/tex]
Hence, the profit for this retailer is:
[tex]\[ \$30 \][/tex]
To summarize:
- Cost equation: [tex]\( y = 100 + 2x \)[/tex]
- Revenue equation: [tex]\( y = 15x \)[/tex]
- Total costs when 10 items are made: [tex]\( \$120 \)[/tex]
- Total revenue from selling 10 items: [tex]\( \$150 \)[/tex]
- Profit for the retailer: [tex]\( \$30 \)[/tex]
