- The cost function is:

\[C( x) = 6512 + 17x\]

- The revenue function is:

\[R( x) = 357x - 2x^ 2\]

- The profit function is:

\[P( x) = - 2x^ 2 + 340x - 6512\]

- The smallest break-even point is:

x = 22

UpStudy Free Solution:

Let's go through the problem step-by-step to find the cost function, the revenue function, and the smallest break-even point.

Step 1: Find the cost function

The cost function \(C( x) \) includes the fixed cost and the variable cost. Given that the fixed cost is $6512 and the variable cost per unit (tub of ice cream) is $17, the cost function is:

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

\[C( x) = 6512 + 17x\]

Step 2: Find the revenue function

The revenue function \(R( x) \) is the product of the number of units sold \(x\) and the price-demand function \(p( x) \).

Given the price-demand function:

\[p( x) = 357 - 2x\]

The revenue function is:

\[R( x) = x \cdot p( x) \]

\[R( x) = x ( 357 - 2x) \]

\[R( x) = 357x - 2x^ 2\]

Step 3: Find the profit function

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

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

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

\[P( x) = ( 357x - 2x^ 2) - ( 6512 + 17x) \]

\[P( x) = 357x - 2x^ 2 - 6512 - 17x\]

\[P( x) = - 2x^ 2 + 340x - 6512\]

Step 4: Determine the break-even point

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

\[- 2x^ 2 + 340x - 6512 = 0\]

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

- \(a = - 2\)

- \(b = 340\)

- \(c = - 6512\)

We solve this quadratic equation using the quadratic formula:

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

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

\[x = \frac { - 340 \pm \sqrt { 340^ 2 - 4( - 2) ( - 6512) } } { 2( - 2) } \]

\[x = \frac { - 340 \pm \sqrt { 115600 - 4( 2) ( 6512) } } { - 4} \]

\[x = \frac { - 340 \pm \sqrt { 115600 - 52096} } { - 4} \]

\[x = \frac { - 340 \pm \sqrt { 63504} } { - 4} \]

\[x = \frac { - 340 \pm 252} { - 4} \]

This gives us two solutions:

\[x = \frac { - 340 + 252} { - 4} = \frac { - 88} { - 4} = 22\]

\[x = \frac { - 340 - 252} { - 4} = \frac { - 592} { - 4} = 148\]

So, the break-even points are at \(x = 22\) and \(x = 148\).

Supplemental Knowledge: Understanding these functions' significance in business decision-making is crucial. The cost function represents total production costs at different output levels. The revenue function shows how much money comes from sales at various prices and quantities. The profit function helps businesses determine when they start making a profit after covering all costs.

In business analysis, identifying break-even points helps companies understand how many units they need to sell to cover costs fully. This information aids in pricing strategies, budgeting, and forecasting.

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