Question

unit 5: investing
use the formula
$a = p(1 + \frac{r}{n})^{nt}$
where $a$ represents the total amount, $p$ represents the principal (initial amount invested), $r$ represents the interest rate (as a decimal), $n$ represents the amount of times the account is compounded in one year, and $t$ represents the number of years to answer the following questions.

71.)

a coin had a value of $23.76 in 2005. its value has been increasing at 11% per year. what is the value of the coin after 7 years?

Explanation:

Step1: Identify the values

Here, \( P = 23.76 \) (initial value), \( r = 0.11 \) (11% as a decimal), \( n = 1 \) (compounded annually), and \( t = 7 \) (number of years).

Step2: Substitute into the formula

Substitute the values into the compound - interest formula \( A = P(1+\frac{r}{n})^{nt} \).
\[

$$\begin{align*} A&=23.76\times(1 + \frac{0.11}{1})^{1\times7}\\ &=23.76\times(1 + 0.11)^{7}\\ &=23.76\times(1.11)^{7} \end{align*}$$

\]
First, calculate \( 1.11^{7}\approx1.11\times1.11\times1.11\times1.11\times1.11\times1.11\times1.11 \)
\( 1.11^{2}=1.2321 \), \( 1.11^{3}=1.2321\times1.11 = 1.367631 \), \( 1.11^{4}=1.367631\times1.11=1.51807041 \)
\( 1.11^{5}=1.51807041\times1.11 = 1.6850581551 \), \( 1.11^{6}=1.6850581551\times1.11=1.870414552161 \)
\( 1.11^{7}=1.870414552161\times1.11 = 2.07616015289871 \)
Then, \( A = 23.76\times2.07616015289871\approx23.76\times2.0762 \)
\( 23.76\times2.0762=(20 + 3.76)\times2.0762=20\times2.0762+3.76\times2.0762=41.524+7.806512 = 49.330512\approx49.33 \)

Answer:

The value of the coin after 7 years is approximately \(\$49.33\)