Chapter 3.3, Problem 96AYU

Birthrate for Unmarried Women In the United States, the birthrate $B$ for unmarried women (births per 1000 unmarried women) whose age is $a$ is modeled by the function $B\left(a\right)=-0.30a^2+16.26a-158.90$.

(a) What is the age of unmarried women with the highest birthrate?

(b) What is the highest birthrate of unmarried women?

(c) Evaluate and interpret $B\left(40\right)$.

Source: National Vital Statistics System, 2013

To determine

To calculate:

a. Age of unmarried women with the highest birth rate.

b. Highest birth rate of unmarried women.

c. Evaluate and interpret $B(40)$.

Answer to Problem 96AYU

a. Therefore, the age of unmarried women with the highest birth rate is 27.

b. The highest birth rate of unmarried women is $\text{61}\text{.42}$ years.

c. $B(40)=-0.30{(40)}^2+16.26(40)-158.90=11.5$.

Here, we are asked to find the birth rate of unmarried women of age 40.

Thus, we get that for every 1000 unmarried women, $\text{11}\text{.5}$ of them are of the age 40.

Explanation of Solution

Given:

The relation between the age and the birth rate of unmarried women is

$B(a)=-0.30a^2+16.26a-158.90$

Formula used:

The quadratic equation for finding the value of $x$ in $f(x)=a{x}^2+bx+c,a\ne 0$ is

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Consider a quadratic function of the form $f(x)=a{x}^2+bx+c,a\ne 0$.

Then the graph of the above function is a parabola with vertex $\left(\frac{-b}{2a},f\left(\frac{-b}{2a}\right)\right)$.

This vertex is the highest point if $a<0$ and the lowest point if $a>0$.

Therefore, the maximum (or the minimum) value of the function will be $f\left(\frac{-b}{2a}\right)$.

Calculation:

Here, we have $a=-0.3<0$. Thus, the given function is having the maximum value at the vertex.

Thus, we get

$\frac{-b}{2a}=\frac{-(16.26)}{2(-0.3)}=26.67\approx 27$ And $B\left(\frac{-b}{2a}\right)=B(27)=-0.30{(27)}^2+16.26(27)-158.90=61.42$

Thus, the vertex is at $(27,61.42)$.

a. Therefore, the age of unmarried women with the highest birth rate is 27.

b. The highest birth rate of unmarried women is $\text{61}\text{.42}$ years.

c. $B(40)=-0.30{(40)}^2+16.26(40)-158.90=11.5$.

Here, we are asked to find the birth rate of unmarried women of age 40.

Thus, we get that for every 1000 unmarried women, $\text{11}\text{.5}$ of them are of the age 40.