Chapter 4.1, Problem 44PPS

a.

To determine

To find: Function $d\left(t\right)$ .

Answer to Problem 44PPS

Amount of money in account after “t” weeks = $d(t)=5000+100t$

Explanation of Solution

Given information:

Initial deposit $=\$5000$

Amount Amy deposits every week $=\$100$

Amount of money in account after “t” weeks $=5000+\left(100+100+100+\mathrm{......}"t"times\right)$

  $=5000+\left(100\times t\right)$

  $=5000+100t$

Amount of money in account after “t” weeks = $d(t)=5000+100t$ .

b.

To determine

To find: Function $w\left(t\right)$ .

Answer to Problem 44PPS

Total amount of money withdrawn after $t$ weeks = $w\left(t\right)$

  $=400+250t$

Explanation of Solution

Given information:

Amount of money Amy withdrawn first week $\left(t=1\right)$ = $\$650$

Amount of money withdrawn in following weeks $\left(t=2,3,\mathrm{..}\right)$ = $\$250$

Total amount of money withdrawn = $650+\left(250+250+250+\mathrm{.....}"\left(t-1\right)"times\right)$

  $=650+\left(250\times \left(t-1\right)\right)$

  $=650+250t-250$

  $=400+250t$

Therefore

Total amount of money withdrawn after $t$ weeks = $w\left(t\right)$

  $=400+250t$ .

c.

To determine

To find: Function $B\left(t\right)$ .

Answer to Problem 44PPS

  $B\left(t\right)=4600-150t$

Explanation of Solution

Given information:

Initial deposit $=\$5000$

Amount Amy deposits every week $=\$100$

Amount of money Amy withdrawn first week $\left(t=1\right)$ = $\$650$

Amount of money withdrawn in following weeks $\left(t=2,3,\mathrm{..}\right)$ = $\$250$

  $\begin{array}{l}B\left(t\right)=d\left(t\right)-w\left(t\right)\\ \Rightarrow B\left(t\right)=5000+100t-\left(400+250t\right)\\ \Rightarrow B\left(t\right)=5000+100t-400-250t\\ \Rightarrow B\left(t\right)=4600-150t\end{array}$

Therefore,

  $B\left(t\right)=4600-150t$

d.

To determine

To find: When Amy will run out of money.

Answer to Problem 44PPS

Amy will run out of money After $t=30.66$ weeks.

Explanation of Solution

Given information:

Initial deposit $=\$5000$

Amount Amy deposits every week $=\$100$

Amount of money Amy withdrawn first week $\left(t=1\right)$ = $\$650$

Amount of money withdrawn in following weeks $\left(t=2,3,\mathrm{..}\right)$ = $\$250$

When $B\left(t\right)$ is less than zero then Amy will run out of money.

  $\begin{array}{l}B\left(t\right)<0\\ \Rightarrow 4600-150t<0\\ \Rightarrow 4600<150t\\ \Rightarrow 150t>4600\\ \Rightarrow t>\frac{4600}{150}\\ \Rightarrow t>30.66\end{array}$

Yes, Amy will run out of money After $t=30.66$ weeks.