Question

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 \(B(40)\)

Short answer

The age with the highest birth rate is 29 years. The highest birth rate is 65.521. For 40 years old, the birth rate is 25.43.

Step-by-step solution

Identify the type of function

The given function is a quadratic function of the form B(a) = -0.33a^{2} + 19.17a - 213.37 . Quadratic functions have a parabolic shape and can open upwards or downwards. In this case, the quadratic term has a negative coefficient (-0.33), which means the parabola opens downwards, indicating a maximum point.

Find the critical point (vertex) for part (a)

The vertex of a quadratic function y = ax^{2} + bx + c is found using the formula x = -b / (2a) . Here, a = -0.33 and b = 19.17. Using the formula: a= -0.33 b= 19.17 a= x = -19.17 / (2*(-0.33)) x = 19.17 / 0.66 x = 29.045 . Thus, the age of unmarried women with the highest birth rate is approximately 29 years.

Calculate the maximum birth rate for part (b)

Substitute a = 29.045 into the original function to find the maximum birth rate: B(29.045) = -0.33(29.045)^{2} + 19.17(29.045) - 213.37 ; Using a calculator: B(29.045) = -0.33(843.611) + 557.282 - 213.37 B(29.045)= -278.391 + 557.282 - 213.37 B(29.045) = 65.521 . Thus, the highest birth rate is approximately 65.521 births per 1000 unmarried women.

Evaluate and interpret B(40) for part (c)

Substitute a = 40 into the function to evaluate B(40): B(40) = -0.33(40)^{2} + 19.17(40) - 213.37 ; Simplifying this, B(40) = -0.33(1600) + 766.8 - 213.37 -528 + 766.8 - 213.37 B(40) = 25.43 . This means that the birth rate for unmarried women who are 40 years old is approximately 25.43 births per 1000 unmarried women.

Key concepts

vertex of a parabola

Understanding the vertex of a parabola is crucial when dealing with quadratic functions. The vertex is the point at which the parabola changes direction. If it opens upwards, the vertex is the minimum point. If it opens downwards, the vertex is the maximum point. In the function given in the exercise, we identify that the parabola opens downwards because the coefficient of the quadratic term (-0.33a^2) is negative.

To find the vertex, we use the formula:
x = \(-\frac{b}{2a}\). In this problem, a = -0.33 and b = 19.17. Plugging in these values, we get:

a = -0.33
b = 19.17
x = \(-\frac{19.17}{2*(-0.33)}\) = \( \frac{19.17}{0.66}\) = 29.045

So, the age of unmarried women with the highest birth rate is approximately 29 years. This age corresponds to the vertex of the parabola in this context.

maximum and minimum points

In quadratic functions, determining maximum and minimum points helps us understand crucial points about the function's behavior. For the given quadratic function:
B(a) = -0.33a^2 + 19.17a - 213.37
we have already established that the parabola opens downwards, indicating a maximum point at the vertex.

To find this maximum value, substitute the vertex (the age already found, 29.045) back into the function:

a = 29.045B(29.045) = -0.33(29.045)^2 + 19.17(29.045) - 213.37Using a calculator for accuracy:
= -0.33(843.611) + 557.282 - 213.37
= -278.391 + 557.282 - 213.37
= 65.521
Hence, the highest birth rate for unmarried women is approximately 65.521 births per 1000 unmarried women.

evaluating quadratic functions

Evaluating quadratic functions involves substituting a specific value for the variable in the equation to find the corresponding result. For example, to evaluate and interpreta = 40in the given problem:
B(a) = -0.33a^2 + 19.17a - 213.37

Substitute 40 for a:
B(40) = -0.33(40)^2 + 19.17(40) - 213.37

Simplify step by step:= -0.33(1600) + 766.8 - 213.37
= -528 + 766.8 - 213.37
= 25.43

So, the birth rate for unmarried women who are 40 years old is approximately 25.43 births per 1000 unmarried women. This process shows the application of quadratic functions in real-life scenarios, providing insight into how different ages impact birth rates among unmarried women.