Q. 94 - Precalculus Enhanced with Graphing Utilities

Question

In the United States, the birthrate B of unmarried women (births per 1000 unmarried women) for women whose age is a is modeled by the function

$B\left(a\right) = -0.27a^2 + 14.23a - 120.16$

(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(40).

Step-by-Step Solution

Answer

a)the age of unmarried women with the highest birthrate is $26.35yrs$.

b)There are $67.3$births per 1000 unmarried woman

c) There are $17.03$ births per 1000 unmarried woman at the age of 40

Part a) Step 1: Given information

We are given a quadratic equation $B\left(a\right)=-0.27a^2+14.23a-120.16$

Part a) Step 2: Find the vertex

As the coefficient of $x^2$ is negative the parabola opens downwards.

And to find the highest birthrate of married woman can be found at vertex of the parabola

vertex can be given as 

$$\begin{gathered} x=-\frac{b}{2a} \\ x=\frac{14.23}{0.54} \\ x=26.35yrs \end{gathered}$$

Part b) Step 1: Find the highest birthrate of woman per 1000 woman

TO find the highest birthrate we substitute $x=26.35$ in the equation

We get,

$$\begin{gathered} B(26.35)=-0.27{(26.35)}^2 + 14.23(26.35) - 120.16 \\ B(26.35)=67.32 \end{gathered}$$

Part c) Step 1) Find B ( 40 )

To find $B\left(40\right)$ we substitute $a=40$ in the equation we get,

$$\begin{gathered} B\left(40\right)=-0.27{\left(40\right)}^2+ 14.23\left(40\right) - 120.16 \\ B\left(40\right)=17.06 \end{gathered}$$

There are $17.06$ births per unmarried woman for the age 40years.

Step 5: Conclusion

a) the age of unmarried women with the highest birthrate is $26.35yrs$

b)There are $67.32$ births per $1000$ unmarried woman

c)There are $17.06$ births per 1000 unmarried woman for the age 40years.