Problem 30 - Elementary and Intermediate Algebra

Question

Solve Martina's Custom Printing is planning to add painter's caps to its product line. For the first year, the fixed costs for setting up production are \(\$ 16,404 .\) The variable costs for producing a dozen caps are \(\$ 6.00 .\) The revenue on each dozen caps will be \(\$ 18.00 .\) Find the following. a) The total cost \(C(x)\) of producing \(x\) dozen caps b) The total revenue \(R(x)\) from the sale of \(x\) dozen caps c) The total profit \(P(x)\) from the production and sale of \(x\) dozen caps d) The profit or loss from the production and sale of 3000 dozen caps; of 1000 dozen caps e) The break-even point

Step-by-Step Solution

Answer

a: \(C(x) = 16,404 + 6x\), b: \(R(x) = 18x\), c: \(P(x) = 12x - 16,404\), d: Profit for 3000 dozen caps: \(\text{\textdollar}19,596\), Loss for 1000 dozen caps: \(\text{\textdollar}4,404\), e: Break-even point: 1,367 dozen caps.

Step 1: Define the total cost function

The total cost function, denoted as \(C(x)\), includes both fixed costs and variable costs. The fixed costs are \(\text{\textdollar}16,404\), and the variable costs are \(\text{\textdollar}6.00\) per dozen caps. Therefore, the total cost function is:\[C(x) = 16,404 + 6x\]

Step 2: Define the total revenue function

The total revenue function, denoted as \(R(x)\), is the product of the number of dozen caps sold, \(x\), and the revenue per dozen caps, \(\text{\textdollar}18.00\). Therefore, the total revenue function is:\[R(x) = 18x\]

Step 3: Define the total profit function

The profit function, denoted as \(P(x)\), is the difference between the total revenue and the total cost. So, the profit function is:\[P(x) = R(x) - C(x)\]Substitute the expressions from the previous steps:\[P(x) = 18x - (16,404 + 6x)\]Simplify to get:\[P(x) = 12x - 16,404\]

Step 4: Calculate the profit or loss for 3000 dozen caps

To find the profit or loss for 3000 dozen caps, substitute \(x = 3000\) into the profit function \(P(x)\):\[P(3000) = 12(3000) - 16,404\]Calculate:\[P(3000) = 36,000 - 16,404 = 19,596\]Thus, the profit for 3000 dozen caps is \(\text{\textdollar}19,596\)

Step 5: Calculate the profit or loss for 1000 dozen caps

To find the profit or loss for 1000 dozen caps, substitute \(x = 1000\) into the profit function \(P(x)\):\[P(1000) = 12(1000) - 16,404\]Calculate:\[P(1000) = 12,000 - 16,404 = -4,404\]Thus, the loss for 1000 dozen caps is \(\text{\textdollar}4,404\)

Step 6: Find the break-even point

The break-even point occurs when profit \(P(x)\) is zero. Set the profit function equal to zero and solve for \(x\):\[0 = 12x - 16,404\]Solve for \(x\):\[12x = 16,404\]\[x = \frac{16,404}{12} = 1,367\]Thus, the break-even point is 1,367 dozen caps.

Key Concepts

total cost function

Martina's Custom Printing has fixed costs and variable costs for producing painter's caps.
Fixed costs are expenses that remain the same regardless of the number of caps produced.
In this case, the fixed costs amount to \text{\textdollar}16,404\.

Variable costs change based on the production level.
Here, it costs \text{\textdollar}6.00\ to produce a dozen caps.
The total cost function, denoted by \textbf{C(x)}\, combines both these costs.

The formula is:
\[C(x) = 16,404 + 6x\]
Where:
\begin{itemize}
\text{\textdollar}16,404\: fixed costs
x: number of dozen caps
\text{\textdollar}6.00\: variable cost per dozen caps

Understanding this helps you know how costs behave with different production levels.

total revenue function

Revenue is the total income from selling goods.
Martina's Custom Printing makes \text{\textdollar}18.00\ for each dozen caps sold.

The total revenue function, denoted by \textbf{R(x)}\, shows how much money comes in based on the number of dozens sold.

Use the formula:
\[R(x) = 18x\]
Where:
\begin{itemize}
x: number of dozen caps sold
\text{\textdollar}18.00\: revenue per dozen caps

By understanding this, you can predict earnings based on sales.

profit function

Profit is what's left after subtracting costs from revenue.
Martina's Custom Printing calculates profit by using the total profit function \textbf{P(x)}\.

The formula is:
\[P(x) = R(x) - C(x)\]
This means subtracting the total cost from total revenue.

Substitute the cost and revenue formulas:
\[P(x) = 18x - (16,404 + 6x) = 12x - 16,404\]

This simplified formula shows how profit changes with the number of dozen caps sold.

break-even point

The break-even point is where total revenue equals total costs.
Martina's Custom Printing finds this point by setting the profit function to zero:

Use the formula:
\[0 = 12x - 16,404\]
Solve for x:
\[12x = 16,404 -> x = \frac{16,404}{12} = 1,367\]

This means they need to sell 1,367 dozen caps to cover all costs without any profit or loss.
Understanding the break-even point helps in deciding how much needs to be sold to avoid losses.