D1.02: Examples 2–5

Example 2

If we invest $750 at 6% annual interest, compounded quarterly, the formula for the amount A that the investment is worth after t years is [latex]A=750\cdot{{\left(1+\frac{0.06}{4}\right)}^{4t}}[/latex]. Find the amount the investment is worth after five years. Then evaluate this for several values of t between 0 and 30 years, and sketch a graph of this formula.

Example 3

Consider our approximation from Example 2. We wanted to find the number of years to leave the money in so that the amount of the investment would be $4000. The graph suggested that [latex]t=27.5[/latex] years. But then we found that after 27.5 years, the amount was only $3857.68. Clearly we should leave the money in somewhat longer.  How much longer?

Example 4

Following Example 2, we’d like to have a formula that allows us to vary the interest rate and number of times per year it is compounded as well as varying the number of years of the investment. So let r be the interest rate, converted to a decimal and n be the number of times per year it is compounded, and, as before t is the number of years and A is the amount of the investment.    Then the formula is [latex]A=750\cdot{{\left(1+\frac{r}{n}\right)}^{n\cdot{t}}}[/latex]. This is an example of a formula with several input values. Use this formula to find the amount of the investment after 5 years if the interest rate is 8% and it is compounded monthly.

Example 5

Following Example 3, suppose we want to allow the initial amount of the investment to change. So we need a variable for that. Since it is an amount, we’d like to call it A, but we already have an A in this formula that means something else. We could use a different letter, but in applications problems we often choose to call both values A and distinguish between them by a subscript. In this problem, we would usually call the original amount of money [latex]{{A}_{0}}[/latex] and the final amount of money after t years [latex]{{A}_{t}}[/latex]. So the formula is   [latex]{{A}_{t}}={{A}_{0}}\cdot{{\left(1+\frac{r}{n}\right)}^{n\cdot{t}}}[/latex]. Use this formula to find the amount of an investment after 6 years if the initial amount is $900, the annual rate is 0.07 and it is compounded twice a year.