Chapter 4.6, Problem 41PPS

a.

To determine

Find the number of death per $100,000$ people for year $2010and2015$

Answer to Problem 41PPS

  $2010is60.31and2015is25.06$

Explanation of Solution

Given information:

The number of deaths per $100,000$ people $y$ can be approximated by $y=-0.26x^2-0.55x+91.81$ , where $x$ represents the number of years after $2000$ .

  $\begin{array}{cc}Year& \begin{array}{l}Deaths,per\\ 100,000\end{array}\\ 2000& 91.9\\ 2002& 89.4\\ 2004& 85.3\\ 2010& ?\\ 2015& ?\end{array}$

Calculation:

Consdier the equation

  $y=-0.26x^2-0.55x+91.81$

Where $x$ represents the number of years after $2000$ .

The number of deaths per $100,000$ people for $2010$ is calculated below.

  $\begin{array}{l}y=-0.26x^2-0.55x+91.81\\ =-0.26{(}^{10}-0.55(10)+91.81\\ =60.31\end{array}$

Hence, the number of deaths in $2010is60.31$ .

The number of deaths per $100,000$ people for $2015$ is calculated below.

  $\begin{array}{l}y=-0.26x^2-0.55x+91.81\\ =-0.26{(}^{15}-0.55(15)+91.81\\ =25.06\end{array}$

Hence, the number of deaths in $2015is25.06$ .

b.

To determine

Find $x$ by quadratic formula when, $y=50$ .

Answer to Problem 41PPS

  $11.58$

Explanation of Solution

Given information:

The number of deaths per $100,000$ people $y$ can be approximated by $y=-0.26x^2-0.55x+91.81$ , where $x$ represents the number of years after $2000$ .

  $\begin{array}{cc}Year& \begin{array}{l}Deaths,per\\ 100,000\end{array}\\ 2000& 91.9\\ 2002& 89.4\\ 2004& 85.3\\ 2010& ?\\ 2015& ?\end{array}$

Calculation:

Consdier the equation

  $y=-0.26x^2-0.55x+91.81$

Where $x$ represents the number of years after $2000$ .

The value is calculated below.

  $\begin{array}{l}y=-0.26x^2-0.55x+91.81\\ 50=-0.26x^2-0.55x+91.81\\ 0=-0.26x^2-0.55x+41.81\end{array}$

Therefore,

  $\begin{array}{l}x=\frac{-\left(-0.55\right)\pm \sqrt{{\left(-0.55\right)}^2-4\left(-0.26\right)\left(41.81\right)}}{2\left(-0.26\right)}\\ =\frac{0.55\pm \sqrt{43.7894}}{-0.52}\\ x\approx -13.8or11.7\end{array}$

Hence, the number of years after $2000$ cannot be negative, the solution is $11.58$ .

Hence, $11.7$ years after $2000$ , or in $2011$ , the death rate will be $50per100,000$ .

c.

To determine

Predict the year when death rate for per $100,000$ people will be zero.

Answer to Problem 41PPS

  $2017$

Explanation of Solution

Given information:

The number of deaths per $100,000$ people $y$ can be approximated by $y=-0.26x^2-0.55x+91.81$ , where $x$ represents the number of years after $2000$ .

  $\begin{array}{cc}Year& \begin{array}{l}Deaths,per\\ 100,000\end{array}\\ 2000& 91.9\\ 2002& 89.4\\ 2004& 85.3\\ 2010& ?\\ 2015& ?\end{array}$

Calculation:

Consdier the equation

  $y=-0.26x^2-0.55x+91.81$

Where $x$ represents the number of years after $2000$ .

The value is calculated below.

  $\begin{array}{l}y=-0.26x^2-0.55x+91.81\\ 0=-0.26x^2-0.55x+41.81\end{array}$

Therefore,

  $\begin{array}{l}x=\frac{-\left(-0.55\right)\pm \sqrt{{\left(-0.55\right)}^2-4\left(-0.26\right)\left(91.81\right)}}{2\left(-0.26\right)}\\ =\frac{0.55\pm \sqrt{95.7894}}{-0.52}\\ x\approx -19.88or17.76\end{array}$

Hence, the number of years after $2000$ cannot be negative, the solution is $11.76$ .

Hence, $11.76$ years after $2000$ , or in $2017$ , the death rate will be $0per100,000$ .