Questions: The unit cost, in dollars, to produce bins of cat food is 19 and the fixed cost is 9408. The price-demand function, in dollars per bin, is (p(x)=311-2x) Find the cost function. C(x)= Find the revenue function. R(x)= Find the profit function. P(x)= At what quantity is the smallest break-even point? Select an answer

Find the cost function. Define the cost function The cost function is given by \(C(x) = \text{variable cost} + \text{fixed cost}\). The variable cost is the unit cost multiplied by the quantity produced, and the fixed cost is given as \(\$9408\). The unit cost is \(\$19\) per bin. Therefore, the variable cost is \(19x\). The cost function is given by \(C(x) = 19x + 9408\).

\(C(x) = 19x + 9408\)

Find the revenue function. Define the revenue function The revenue function is given by \(R(x) = p(x) \cdot x\), where \(p(x)\) is the price-demand function and \(x\) is the quantity. Given that \(p(x) = 311 - 2x\), the revenue function is \(R(x) = (311 - 2x)x = 311x - 2x^2\).

\(R(x) = 311x - 2x^2\)

Find the profit function. Define the profit function The profit function is given by \(P(x) = R(x) - C(x)\), where \(R(x)\) is the revenue function and \(C(x)\) is the cost function. Given \(R(x) = 311x - 2x^2\) and \(C(x) = 19x + 9408\), we have \(P(x) = (311x - 2x^2) - (19x + 9408) = 311x - 2x^2 - 19x - 9408 = -2x^2 + 292x - 9408\).

\(P(x) = -2x^2 + 292x - 9408\)

At what quantity is the smallest break-even point? Define break-even point A break-even point occurs when the revenue equals the cost, i.e., \(R(x) = C(x)\) or \(P(x) = 0\). We need to solve the equation \(P(x) = -2x^2 + 292x - 9408 = 0\). Solve for \(x\) Dividing by \(-2\), we get \(x^2 - 146x + 4704 = 0\). We can use the quadratic formula to solve for \(x\): \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] where \(a=1\), \(b=-146\), and \(c=4704\). \[x = \frac{146 \pm \sqrt{(-146)^2 - 4(1)(4704)}}{2(1)} = \frac{146 \pm \sqrt{21316 - 18816}}{2} = \frac{146 \pm \sqrt{2500}}{2} = \frac{146 \pm 50}{2}\] So, \(x = \frac{146 + 50}{2} = \frac{196}{2} = 98\) and \(x = \frac{146 - 50}{2} = \frac{96}{2} = 48\).

The smallest break-even point is \(x = 48\).

\(48\)

\(C(x) = 19x + 9408\) \(R(x) = 311x - 2x^2\) \(P(x) = -2x^2 + 292x - 9408\) \(48\)