In a workbook for our AP Calculus class, I found this question:
The Sales of a new product, after it has been on the market for $t$ years, is given by
$S(t)=Ce^{k/t}$
a) Find C and k if 7000 units are sold after one year and $\lim _{t\to\infty }S(t)=45,000$
b) Find the total number of units sold during the year t=5 and t=10.
Part (a) was easy to do when you apply the limits and set 7000 equal to S(t); the values of C and k are 45,000 and $ln(7/45)$ respectively;
Part b is the portion I am unable to answer. It tried plugging in those values and then adding them together but that is incorrect; also, my teacher said it had to do with integration but integrating the S(t) does not give you the units sold. It gives Unit multiplied by time when you find the accumulated value under the curve of S(t). I have not been able to figure it out. How would I go about figuring this out using some form of integration?
$S(x)$ is the total mumber of sold units until $t=x$. Therefore the total number of sold units between $t=a$ and $t=b$ is $S(b)-S(a)$. Thus the total number of sold units between $t=5$ and $t=10$ is
$$S(10)-S(5)=45,000\cdot \left( \frac7{45} \right)^{\frac1{10}}-45,000\cdot \left( \frac7{45} \right)^{\frac1{5}}$$
Remark
$e^{\ln (7/45)\cdot 1/t}$ can be transformed to $$\left(e^{\ln (7/45)}\right)^{\frac1t}\Rightarrow \left( \frac7{45} \right)^{\frac1{t}}$$