Birth Rate of Unmarried Women In the United States, the...

Birth Rate 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 $B(a)=-0.30 a^{2}+16.26 a-158.90$ (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 $B(40) .$

Birth Rate 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 $B(a)=-0.30 a^{2}+16.26 a-158.90$
(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

Vertex of a Parabola

The vertex of a parabola is the point where the function attains its maximum or minimum value. For quadratic functions, the vertex can be found using the formula x = -b/(2a). This concept is crucial for optimization since it enables one to identify the point at which a function reaches its peak or lowest value, which is particularly useful in real-world applications such as determining optimal performance or outcomes.

Quadratic Function

A quadratic function is a second-degree polynomial expressed in the form f(x) = ax^2 + bx + c, where a, b, and c are constants and a is nonzero. Its graph is a parabola, which opens upward if a is positive and downward if a is negative. This type of function is commonly used to model situations that involve acceleration or areas, and it provides a simple yet powerful way of understanding relationships between variables.

Function Evaluation and Interpretation

Function evaluation involves substituting a specific value into a function to calculate the corresponding output. Interpreting the result within the context of the modeled situation allows one to understand the real-world implications of the mathematical solution. This process connects the abstract mathematical model to practical scenarios by providing meaningful numerical insights.

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