In the United States, the birth rate B of unmarried women...

In the United States, the birth rate $B$ of unmarried women (births per 1000 unmarried women) for women whose age is $a$ is modeled by the function $B(a)=-0.33 a^{2}+19.17 a-213.37$ (a) What is the age of unmarried women with the highest birth rate? (b) What is the highest birth rate of unmarried women? (c) Evaluate and interpret $B(40)$

In the United States, the birth rate $B$ of unmarried women (births per 1000 unmarried women) for women whose age is $a$ is modeled by the function $B(a)=-0.33 a^{2}+19.17 a-213.37$
(a) What is the age of unmarried women with the highest birth rate?
(b) What is the highest birth rate of unmarried women?
(c) Evaluate and interpret $B(40)$

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Key Concepts

Quadratic Functions

Quadratic functions are second-degree polynomial functions typically written in the form f(x) = ax^2 + bx + c. They produce parabolic graphs that are either concave up or down based on the sign of the coefficient a, and are commonly used to model various real-world phenomena, including demographic rates, physics problems, and economic behaviors.

Vertex of a Parabola

The vertex of a parabola is the point where the quadratic function reaches its maximum or minimum value. It is a crucial concept in optimization problems and can be found using the formula x = -b/(2a) when the function is in standard form. This concept allows one to determine the peak or bottom point in the context of the modeled scenario, such as the age corresponding to the highest birth rate.

Function Evaluation in Modeling

Evaluating a function at a specific point is the process of substituting a value into the function to obtain a result. In real-world modeling, such as predicting birth rates based on age, this approach helps to interpret the model's implications by providing a numerical prediction at a given data point. It is critical for understanding and applying the insights of the function to practical scenarios.

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