Problem 98 - A Graphical Approach to Precalculus with Limits

Question

For each simation, if \(x\) represents the number of items produced, (a) write a cost function, (b) find a revenue function if each item sells for the price given, (c) state the profit function, (d) determine analyrically how many ilems must be produced before a profit is realized (assume whole numbers of items), and (e) support the result of part (d) graphically. The fixed cost is \(\$ 1000,\) the cost to produce an item is \(\$ 200,\) and the selling price of the item is \(\$ 240\).

Step-by-Step Solution

Answer

Profit is realized when producing at least 26 items.

Step 1: Write the Cost Function

The cost function is the sum of fixed costs and variable costs. The fixed cost is \(1000, and the variable cost per item is \)200. Therefore, the cost function is: \[ C(x) = 1000 + 200x \] where \( x \) is the number of items produced.

Step 2: Find the Revenue Function

The revenue function calculates the total income from selling \( x \) items, where each item sells for $240. Thus, the revenue function is: \[ R(x) = 240x \] where \( x \) is the number of items sold.

Step 3: State the Profit Function

The profit function is the revenue function minus the cost function. It is defined as: \[ P(x) = R(x) - C(x) \] Substitute the previously found functions: \[ P(x) = 240x - (1000 + 200x) \] which simplifies to: \[ P(x) = 40x - 1000 \]

Step 4: Determine the Break-even Point Analytically

To find out how many items must be produced for a profit, we set the profit function greater than zero: \[ 40x - 1000 > 0 \] Solve for \( x \): \[ 40x > 1000 \] \[ x > 25 \] Since only whole items can be produced, a profit is realized when \( x = 26 \).

Step 5: Graphically Support the Break-even Point

Graph both the cost function \( C(x) = 1000 + 200x \) and the revenue function \( R(x) = 240x \). The break-even point occurs where the revenue line crosses above the cost line. Verify graphically that for \( x = 26 \), revenue exceeds costs, confirming that 26 items must be produced to realize a profit.

Key Concepts

Cost Function

The cost function helps us understand how much it costs to produce a certain number of items. In this scenario, the fixed cost, which is the expense that does not change regardless of the number of items produced, is \(\\(1000\). Additionally, the variable cost per item, which varies with the number of units produced, is \(\\)200\) per item.

Putting these together, the cost function can be expressed as:

• Fixed Cost: \(1000\)
• Variable Cost: \(200x\), where \(x\) is the number of items produced
• Cost Function Formula: \(C(x) = 1000 + 200x\)

This equation tells us that to calculate the total cost \(C(x)\), you start with your fixed costs and then add \(200\) for every item \(x\) produced.

Revenue Function

The revenue function is a crucial financial tool used to determine how much money comes in from selling \(x\) items. In our case, each item is sold for \(\$240\), which means that the revenue increases by this amount with each additional unit sold.

This relationship is simple and linear:

• Price per Item: \(240\)
• Revenue Function Formula: \(R(x) = 240x\)

Using the formula, you can easily find out how much total revenue you will earn by multiplying \(240\) by the number of items \(x\) you sell. This equation clearly shows that total income from sales grows with each item sold.

Profit Function

The profit function combines both the cost and revenue functions to determine the actual financial gain from selling \(x\) items. It tells us the difference between the revenue and the total costs.

To compute profit, we use:

• Profit Function: \(P(x) = R(x) - C(x)\)
• Using Given Functions: \(P(x) = 240x - (1000 + 200x)\)
• Simplified Profit Function: \(P(x) = 40x - 1000\)

This shows that profit starts negatively due to fixed costs, but as more items are sold, the profit increases by \(\$40\) per item. Understanding this helps businesses assess how many units need to be sold to overcome initial costs and start making a profit.

Break-even Point

The break-even point is a critical financial metric indicating when total revenue equals total costs, resulting in zero profit. It's important to determine how many items need to be sold to cover all expenses.

To find this point, set the profit function greater than zero:

• Equation: \(40x - 1000 > 0\)
• Solve for \(x\): \(x > 25\)
• Whole Items: Since only whole items can be produced: \(x = 26\)

Thus, producing and selling 26 items will ensure that not only are all costs covered, but also that profit is beginning to be realized beyond the break-even point.