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 a. Therefore, the age of unmarried women with the highest birth rate is 27. b. The highest birth rate of unmarried women is 61 .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, 11 .5 of them are of the age 40. Explanation Given: The relation between the age and the birth rate of unmarried women is B ( a ) = − 0.30 a 2 + 16.26 a − 158.90 Formula used: The quadratic equation for finding the value of x in f ( x ) = a x 2 + b x + c , a ≠ 0 is x = − b ± b 2 − 4 a c 2 a Consider a quadratic function of the form f ( x ) = a x 2 + b x + c , a ≠ 0 . Then the graph of the above function is a parabola with vertex ( − b 2 a , f ( − b 2 a ) ) . 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 ( − b 2 a ) . Calculation: Here, we have a = − 0.3 < 0 . Thus, the given function is having the maximum value at the vertex. Thus, we get − b 2 a = − ( 16.26 ) 2 ( − 0.3 ) = 26.67 ≈ 27 And B ( − b 2 a ) = 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 61 .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, 11 .5 of them are of the age 40.

Precalculus Enhanced with Graphing Utilities

6th Edition

ISBN: 9780321795465

Author: Michael Sullivan, Michael III Sullivan

Publisher: PEARSON

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Textbook Question

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

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

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

Source: National Vital Statistics System, 2013

Expert Solution & Answer

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 $61 .42$ 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, $11 .5$ of them are of the age 40.

Explanation of Solution

Given:

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

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

Formula used:

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

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

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

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

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 $f (\frac{- b}{2 a})$ .

Calculation:

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

Thus, we get

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

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 $61 .42$ 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, $11 .5$ of them are of the age 40.

Chapter 3.3, Problem 95AYU

Chapter 3.3, Problem 97AYU

Chapter 3 Solutions

Precalculus Enhanced with Graphing Utilities

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Solution Summary: The author analyzes the relationship between the age and the birth rate of unmarried women. The graph of the given function is a parabola with vertex.