Question

The chief financial officer of a company reports that profits for the past fiscal year were $\mathrm{\$}15.9$ million. The officer predicts that profits for the next $5years$ will grow at a continuous annual rate somewhere between $3\frac{1}{2}\mathrm{\%}$ and $5\mathrm{\%}.$ Estimate the cumulative difference in total profit over the $5years$ based on the predicted range of growth rates.

Answer 1

Estimating cumulative difference in total profit:

Given, Profit for the past fiscal year $=\mathrm{\$}15.9$ million.

From the data given, we know that the bank profit can vary form

$0.035\times 15900000+15900000=16456500$

and

$0.05\times 15900000+15900000=16695000$

Since the profit will grow continuously the growth can be described as linear function.

So, the profit the total difference can be interpreted as,

$$\begin{gathered} ={\int }_0^5\left(16695000x-16456500x\right)dx \\ ={\int }_0^5\left(238500x\right)dx \\ ={\left[238500\frac{x^2}{2}\right]}_0^5 \\ =2981250 \end{gathered}$$

Hence, cumulative difference in profit is $2981250$.