The annual sales of a new company are expected to grow at a rate proportional to the difference between the sales at time $t$ and an upper limit of $$5$ million. Suppose the sales are $0 initially and are $$2$ million after $4$ years of operation. Determine the annual sales at any time $t$. How long will it take for the annual sales to reach $$4$ million?

Solution Since there are no sales initially ($R=0$), we use equation (4). Then since the upper limit of sales is $$5$ million, the sales at time $t$ are $$\begin{equation*} y(t)=5(1-e^{-kt})\qquad \color{#0066A7}{{M=5}} \end{equation*}$$

Since sales are $$2$ million after $4$ years, the boundary condition is $y(4)=2.$ We use the boundary condition to find the constant $k$: $$\begin{eqnarray*} 2& =&5(1-e^{-4k})\qquad \color{#0066A7}{{y(4)=2}}\\ 5e^{-4k}& =&3 \\[2pt] e^{-4k}& =&0.6 \\[4pt] k& =&\frac{\ln 0.6}{-4} = -0.25\ln 0.6 \end{eqnarray*}$$

So, the function $$\begin{equation*} y(t)=5\big(1-e^{( 0.25\ln 0.6) t}\big) \end{equation*}$$

models the company's sales in year $t.$

To find out how long it will take for sales to reach $$4$ million, we solve the equation $y( t) =4.$ Then $$\begin{eqnarray*} 4& =&5\big( 1-e^{( 0.25\ln 0.6) t}\big) \\ 5e^{( 0.25\ln 0.6) t}& =&1 \\[1pt] (0.25\ln 0.6) t & =&\ln 0.2 \\[3pt] t& =&\frac{\ln 0.2}{0.25\ln 0.6}\approx 12.6 \hbox{years} \end{eqnarray*}$$

It will take approximately $12.6$ years for annual sales to reach $$4$ million.