The cost to produce bottled spring water is given by C(x)=16...

The cost to produce bottled spring water is given by $C(x)=16 x-63,$ where $x$ is the number of thousands of bottles. The total income (revenue) from the sale of these bottles is given by the function $R(x)=-x^{2}+326 x-7463$. since profit $=$ revenue $-$ cost, the profit function must be $P(x)=-x^{2}+310 x-7400$ (verify). how many bottles sold will produce the maximum profit? What is the maximum profit?

The cost to produce bottled spring water is given by $C(x)=16 x-63,$ where $x$ is the number of thousands of bottles. The total income (revenue) from the sale of these bottles is given by the function $R(x)=-x^{2}+326 x-7463$. since profit $=$ revenue $-$ cost, the profit function must be $P(x)=-x^{2}+310 x-7400$ (verify).
how many bottles sold will produce the maximum profit? What is the maximum profit?

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Key Concepts

Profit Function and Optimization

The profit function represents the difference between total revenue and total cost. In business applications, profit optimization involves finding the quantity that maximizes this function. When the profit function is quadratic, identifying its vertex provides the optimal production or sales level to achieve maximum profit. This process is an application of optimization techniques in economics and business decision-making.

Vertex of a Parabola

The vertex of a parabola is the point where the function attains its maximum or minimum value. For a quadratic function in the form ax² + bx + c, the x-coordinate of the vertex can be found using the formula -b/(2a). This concept is essential in optimization problems because the vertex directly gives the input value at which the optimum occurs and is used to compute the corresponding maximum or minimum output.

Quadratic Function

A quadratic function is a polynomial function of degree two, generally expressed in the form ax² + bx + c. Its graph is a parabola which opens upward if the coefficient a is positive (indicating a minimum point) or downward if a is negative (indicating a maximum point). The quadratic function is central in modeling scenarios where the rate of change is variable and where optimal values (maximum or minimum) are sought.

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