College Algebra, Chapter 3, Review Exercises, Section Review Exercises, Problem 60

Chris is saving for his retirement by making regular deposits into a $410k$ plan. As his salary rises, he finds that he can deposit increasing amounts each year. Between the years $1995$ and $2008$, the annual amount(in dollars) that he deposited was given by the function $D(t) = 3500 + 15 t^2$, where $t$ represents the year of the deposit measured fron the store of the plan.
a.) Find $D(0)$ and $D(15)$. What do these values represent?
b.) At what year will he deposit $\$17,000$? Assuming that his deposits continue to be modeled by the function $D$.
c.) Determine the average rate of change of $D$ between $t = 0$ and $t = 15$. What does this number represent?



$\begin{equation} \begin{aligned} \text{a.) } D(0) &= 3500 + 15 (0) ^2 &&& D(15) &= 3500+15(15)^2\\ \\ &= 3500 &&& &= 6875 \end{aligned} \end{equation}$

$D(0)$ represents the amount of Chris deposit from the start of the plan in 1995. On the other hand, $D(15)$ represents the amount of Chris deposit 15 years after the start of the plan, in 2010.

b.) $D = 17,000$ then

$\begin{equation} \begin{aligned} 17,000 &= 3500+15t^2 && \text{Solve for } t \text{, subtract 3500}\\ \\ 15t^2 &= 13,500 && \text{Divide }15\\ \\ t^2 &= 900 && \text{Take the square root}\\ \\ t &= \pm 30 && \text{Choose } t > 0 \end{aligned} \end{equation}$

It shows that in the year $2025$, Chris will deposit $\$17,000$
c.) Recall that the formula for average rate is
$\displaystyle \frac{f(b) - f(a)}{b-a}$

Thus,
$\displaystyle \frac{D(15) - D(0)}{15 - 0} = \frac{6875-3500}{15} = 225$
This number represents the average rate of increase of Chris deposit between the years 1945 and 2010.