Chapter 12.4, Problem 43E

a.

To determine

Predict the average monthly benefit in 2017.

Answer to Problem 43E

  $\boxed{\$808.62}$

Explanation of Solution

Given information:

The average monthly Social Security benefit (in dollars) for a retired worker age 62 or over from 2001 through 2010 is given by where represents the year, with corresponding to2001.

Use the model to predict the average monthly benefit in 2017.

Calculation:

To find the average monthly benefit in 2017, replace $t$ by $17$ in the function $B=\frac{855.0-11674t+5.565t^2}{1.0-0.16t+0.008t^2}$ and then evaluate.

  $B=\frac{855.0-11674\left(17\right)+5.565{\left(17\right)}^2}{1.0-0.16\left(17\right)+0.008{\left(17\right)}^2}$

  $\begin{array}{l}=\frac{478.705}{0.592}\\ \approx 808.62\end{array}$

Hence, the average monthly benefit in 2017 is about $\boxed{\$808.62}$ .

b.

To determine

What is the limit of the function as $t$ approaches infinity?

Answer to Problem 43E

  $\boxed{\$695.63}$

Explanation of Solution

Given information:

The average monthly Social Security benefit (in dollars) for a retired worker age 62 or over from 2001 through 2010 is given by where represents the year, with corresponding to2001.

What is the limit of the function as $t$ approaches infinity? Explain the meaning of the limit in the context of the problem.

Calculation:

Determine the limit of average monthly benefit function as $t$ approaches infinity, that is, find the limit

  $\underset{t\to \infty }{\lim }\frac{855.0-11674t+5.565t^2}{1.0-0.16t+0.008t^2}$

  $=\underset{t\to \infty }{\lim }\frac{\frac{855.0}{t^2}-\frac{11674}{t}+5.565}{\frac{1.0}{t^2}-\frac{0.16}{t}+0.008}$

  $\begin{array}{l}=\frac{0-0+5.565}{0-0+0.008}\\ =\frac{478.705}{0.592}\\ \approx 695.63\end{array}$

Hence, the limit of average monthly benefit function as $t$ approaches infinity ia about $\boxed{\$695.63}$ which means as the time increases, the average monthly Social Security benefit approaches to $\boxed{\$695.63}$ .

c.

To determine

Answer to Problem 43E

  $not$ realistic because this model lasts at $\$695.63$ .

Explanation of Solution

Given information:

The average monthly Social Security benefit (in dollars) for a retired worker age 62 or over from 2001 through 2010 is given by where represents the year, with corresponding to2001.

Discuss why this model is not realistic for long-term predictions of average monthly Social Security benefits.

Calculation:

As time increases, retired work will need more average monthly Social Security benefit, however this model lasts at $\$695.63$ ,

Hence, the model is $not$ realistic for long-term predictions of average monthly Social Security benefits.