# Suppose that the sales at Borders bookstores went from 78 million dollars in 1990 to 424 million dollars in 1992. How do you find an exponential function to model the sales (in millions of dollars) as a function of years, t, since 1990?

Nov 2, 2017

$y \left(t\right) = \left(78 \cdot {10}^{7}\right) \cdot {e}^{\left(\ln \left(\frac{212}{39}\right)\right) t}$

#### Explanation:

$y \left(t\right) = a \cdot {e}^{k t}$
$\ln {e}^{x} = x$

$y \left(t\right) =$ value (of sales) at a certain time

$=$ sales in $1992 = 4.24 \cdot {10}^{8}$

$a$ = value at the start of a given time period

$=$ sales in $1990 = 7.8 \cdot {10}^{7}$

$t$ = time between starting (year) and a later time given

$= 1992 - 1990 = 2$

substitute these values into the equation to find $k$:

$y \left(t\right) = a \cdot {e}^{k t}$
$4.24 \cdot {10}^{8} = \left(7.8 \cdot {10}^{7}\right) \cdot {e}^{2 k}$

${e}^{2 k} = \frac{212}{39}$

$\ln \left({e}^{2 k}\right) = 2 k = \ln \frac{212}{39}$

$2 k = \ln \left(\frac{212}{39}\right)$

$k = \frac{\ln \left(\frac{212}{39}\right)}{2}$

enter $k$ into the equation:

$y \left(t\right) = a \cdot {e}^{k t}$

$y \left(t\right) = \left(78 \cdot {10}^{7}\right) \cdot {e}^{\ln \left(\frac{212}{39}\right)}$