Answer :
To find out the price at which the guitar factory should sell each guitar in order to break even after selling 140 units, let's break down the process step-by-step:
1. Understand the Cost Function:
The cost of production is given by the function [tex]\( C(x) = 100x + 60000 \)[/tex].
This means the variable cost per guitar is \[tex]$100, and there is a fixed cost of \$[/tex]60,000 regardless of how many guitars are produced.
2. Calculate Total Cost for 140 Units:
To find the cost of producing 140 guitars, substitute [tex]\( x = 140 \)[/tex] into the cost function:
[tex]\[
C(140) = 100 \times 140 + 60000
\][/tex]
Calculating this gives:
[tex]\[
C(140) = 14000 + 60000 = 74000
\][/tex]
Therefore, the total cost to produce 140 guitars is \[tex]$74,000.
3. Determine Break-Even Price:
The break-even point occurs when the total revenue equals the total cost. The revenue is calculated by the number of units sold multiplied by the price per unit. Hence, the price per guitar should be:
\[
\text{Price per guitar} = \frac{\text{Total Cost}}{\text{Units Sold}} = \frac{74000}{140}
\]
Calculating this gives approximately 528.57.
4. Round Up to the Nearest Whole Dollar:
Since prices are typically rounded up to ensure the company doesn't fall short, we need to round up to the next whole number. Thus, the company should charge at least \$[/tex]529 per guitar to break even.
Therefore, the company must charge at least \$529 for each guitar to break even after selling 140 units.
1. Understand the Cost Function:
The cost of production is given by the function [tex]\( C(x) = 100x + 60000 \)[/tex].
This means the variable cost per guitar is \[tex]$100, and there is a fixed cost of \$[/tex]60,000 regardless of how many guitars are produced.
2. Calculate Total Cost for 140 Units:
To find the cost of producing 140 guitars, substitute [tex]\( x = 140 \)[/tex] into the cost function:
[tex]\[
C(140) = 100 \times 140 + 60000
\][/tex]
Calculating this gives:
[tex]\[
C(140) = 14000 + 60000 = 74000
\][/tex]
Therefore, the total cost to produce 140 guitars is \[tex]$74,000.
3. Determine Break-Even Price:
The break-even point occurs when the total revenue equals the total cost. The revenue is calculated by the number of units sold multiplied by the price per unit. Hence, the price per guitar should be:
\[
\text{Price per guitar} = \frac{\text{Total Cost}}{\text{Units Sold}} = \frac{74000}{140}
\]
Calculating this gives approximately 528.57.
4. Round Up to the Nearest Whole Dollar:
Since prices are typically rounded up to ensure the company doesn't fall short, we need to round up to the next whole number. Thus, the company should charge at least \$[/tex]529 per guitar to break even.
Therefore, the company must charge at least \$529 for each guitar to break even after selling 140 units.
