- Cost function: \(C( x) = 13192 + 3x\)

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

- Smallest break-even point: \(x = 68\)

UpStudy Free Solution:

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

Cost Function

The cost function \(C( x) \) consists of the fixed cost and the variable cost per unit. Given:

- Fixed cost: $13192

- Variable cost per unit: $3

The cost function \(C( x) \) can be written as:

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

Profit Function

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

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

The profit function \(P( x) \) is:

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

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

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

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

Break-Even Point

The break-even point occurs when the profit is zero, i.e., \(P( x) = 0\):

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

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

\[a = - 2\]

\[b = 330\]

\[c = - 13192\]

To find the roots of this quadratic equation, we use the quadratic formula:

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

Plugging in the values:

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

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

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

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

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

This gives us two solutions:

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

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

Thus, the smallest break-even point is at \(x = 68\).

Supplemental Knowledge

In algebra, cost, revenue, and profit functions are essential concepts in business mathematics. Here's a brief overview:

1. Cost Function (C(x)): This function represents the total cost of producing \(x\) units of a product. It typically includes fixed costs (costs that do not change with the level of production) and variable costs (costs that vary with the level of production). The general form is:

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

2. Revenue Function (R(x)): This function represents the total revenue generated from selling \(x\) units of a product. It can be expressed as:

\[R( x) = ( \text { Price per Unit} ) \times x\]

In this case, the revenue function is given by a quadratic equation:

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

3. Profit Function (P(x)): This function represents the total profit from producing and selling \(x\) units of a product. It is calculated by subtracting the total cost from the total revenue:

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

4. Break-Even Point: The break-even point is where total revenue equals total cost, meaning there is no profit or loss:

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

Understanding these functions helps businesses determine how many units they need to produce and sell to cover their costs and start making a profit.

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