Chapter 9.6, Problem 41PPS

(a)

To determine

Use the quadratic Formula to determine when 20% of the population will have high − speed Internet.

Answer to Problem 41PPS

20% of the population will have high − speed Internet in 1993 and 2023.

Explanation of Solution

Given:

The percent of U.S. households with high − speed internet h can be estimated by $h=-0.2n^2+7.2n+15$ , where n is the number of years since 1990.

Use the quadratic Formula to determine when 20% of the population will have high − speed Internet.

Concept Used:

Use the quadratic Formula to determine when 20% of the population will have high − speed Internet.

Translate the statement into a quadratic equation: $-0.2n^2+7.2n+15=20$

Calculation:

Write the equation in standard form:

  $\begin{array}{l}-0.2n^2+7.2n+15=20\\ -0.2n^2+7.2n-18.5=0\end{array}$

Now solve for n by using the Quadratic Formula: $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

For this equation: $-0.2n^2+7.2n-18.5=0\Rightarrow$

  $a=-0.2,b=7.2,\text{and}c=-18.5$

  $\begin{array}{l}x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\\ x=\frac{-7.2\pm \sqrt{{(7.2)}^2-4·(-0.2)·(-18}.5)}{2·(-0.2)}=\frac{-7.2\pm \sqrt{51.84-14.8}}{-0.4}=\frac{-7.2\pm \sqrt{37.04}}{-0.4}\\ x=\frac{-7.2+\sqrt{37.04}}{-0.4}\text{or}x=\frac{-7.2-\sqrt{37.04}}{-0.4}\\ x\approx 2.75\text{or}x\approx 33.25\end{array}$

Since n is the number of years since 1990 add the solutions to 1990.

  $1990+3=1993\text{and}1990+33=2023$

Thus, 20% of the population will have high − speed Internet in 1993 and 2023.

(b)

To determine

Explain that this quadratic equation a good model for this information.

Answer to Problem 41PPS

66 % of the population would ever have high − speed Internet.

Explanation of Solution

Given:

The percent of U.S. households with high − speed internet h can be estimated by $h=-0.2n^2+7.2n+15$ , where n is the number of years since 1990.

Is a quadratic equation a good model for this information? Explain.

Concept Used:

A quadratic equation is not a good model for this information.

Calculation:

A quadratic equation is not a good model for this information. The parabola has a maximum at about 66, meaning only 66 % of the population would ever have high − speed Internet.

Graph of the equation: $h=-0.2n^2+7.2n+15$