\(C( x) = 13192 + 3x\)

\(P( x) = - 2x^ 2 + 330x - 13192\)

\(x = 68\) is the smallest break-even point

UpStudy Free Solution:

To solve this problem, we need to find the cost function, the profit function, and determine the smallest break-even point.

Step 1: Find the cost function

The total cost function \(C( x) \) is composed of the fixed cost and the variable cost. Given that the fixed cost is $13,192 and the variable cost is $3 per unit produced, the cost function is:

\[C( x) = \text { Fixed Cost} + ( \text { Unit Cost} \times x) \]

\[C( x) = 13192 + 3x\]

Step 2: Find the profit function

The profit function \(P( x) \) is the difference between the revenue function \(R( x) \) and the cost function \(C( x) \):

\[P( x) = R( x) - C( x) \]

Given the revenue function:

\[R( x) = - 2x^ 2 + 333x\]

Substitute \(R( x) \) and \(C( x) \) into the profit function formula:

\[P( x) = ( - 2x^ 2 + 333x) - ( 13192 + 3x) \]

\[P( x) = - 2x^ 2 + 333x - 13192 - 3x\]

\[P( x) = - 2x^ 2 + 330x - 13192\]

Step 3: Determine the break-even point

The break-even point occurs when the profit is zero. Thus, we set the profit function equal to zero and solve for \(x\):

\[- 2x^ 2 + 330x - 13192 = 0\]

This is a quadratic equation of the form \(ax^ 2 + bx + c = 0\), where:

- \(a = - 2\)

- \(b = 330\)

- \(c = - 13192\)

We can solve this quadratic equation using the quadratic formula:

\[x = \frac { - b \pm \sqrt { b^ 2 - 4ac} } { 2a} \]

Substitute the values of \(a\), \(b\), and \(c\):

\[x = \frac { - 330 \pm \sqrt { 330^ 2 - 4( - 2) ( - 13192) } } { 2( - 2) } \]

\[x = \frac { - 330 \pm \sqrt { 108900 - 4( 2) ( 13192) } } { - 4} \]

\[x = \frac { - 330 \pm \sqrt { 108900 - 105536} } { - 4} \]

\[x = \frac { - 330 \pm \sqrt { 3364} } { - 4} \]

\[x = \frac { - 330 \pm 58} { - 4} \]

This gives us two solutions:

\[x = \frac { - 330 + 58} { - 4} = \frac { - 272} { - 4} = 68\]

\[x = \frac { - 330 - 58} { - 4} = \frac { - 388} { - 4} = 97\]

So, the break-even points are at \(x = 68\) and \(x = 97\).

Final Answers:

- The cost function is:

\[C( x) = 13192 + 3x\]

- The profit function is:

\[P( x) = - 2x^ 2 + 330x - 13192\]

- The smallest break-even point is:

\[x = 68\]

In business mathematics, understanding how to determine the cost function, revenue function, and profit function is crucial for assessing the financial health and viability of a business. Here’s a brief overview of these concepts:

1. Cost Function (C(x)):   

- Fixed Costs: These are costs that do not change with the level of output, such as rent, salaries, and equipment costs.   

- Variable Costs: These costs vary directly with the level of production, such as raw materials and direct labor costs.   

- The cost function is typically expressed as \[C( x) = \text { Fixed Cost} + ( \text { Variable Cost per Unit} \times \text { Quantity Produced} ) \]

2. Revenue Function (R(x)):   

- The revenue function represents the total income generated from selling a certain number of units. It is often given or can be derived from the price per unit and the quantity sold.   

- It can be expressed as \(R( x) = ( \text { Price per Unit} \times \text { Quantity Sold} ) \).

3. Profit Function (P(x)):   

- The profit function is the difference between the revenue function and the cost function.   

- It is expressed as \(P( x) = R( x) - C( x) \).

4. Break-Even Point:   

- This is the point at which total revenue equals total costs, resulting in zero profit.   

- It can be found by solving the equation \(P( x) = 0\).

Mastering the intricacies of cost, revenue, and profit calculations is essential for any budding entrepreneur or business student. With UpStudy, the premier multidisciplinary solving and learning platform, you can dive deep into these concepts and more. Whether you're struggling with understanding fixed and variable costs or finding the break-even point for your business, UpStudy offers step-by-step solutions and expert guidance.