Example 6: An investment of $$ \$ 8200 $$ loses value at a rate of $$ 2 \% $$ per year. Use an exponential function to find the value of the investment after 9 years.

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To find the value of the investment after 9 years, we can use an exponential function.

Let's denote the initial investment as $$ P $$ and the rate of loss as $$ r $$ (in decimal form).

The formula for exponential decay is given by:
$$ V = P \times (1 - r)^t $$

Where:
- $$ V $$ is the value of the investment after $$ t $$ years,
- $$ P $$ is the initial investment,
- $$ r $$ is the rate of loss (in decimal form),
- $$ t $$ is the number of years.

Given:
- Initial investment $$ P = \$8200 $$,
- Rate of loss $$ r = 2\% = 0.02 $$ (in decimal form),
- Number of years $$ t = 9 $$.

Substitute the given values into the formula:
$$ V = 8200 \times (1 - 0.02)^9 $$

Now, we can calculate the value of the investment after 9 years using the exponential function.
Calculate the value by following steps:
- step0: Calculate:
  $$8200\left(1-0.02\right)^{9}$$
- step1: Subtract the numbers:
  $$8200\times 0.98^{9}$$
- step2: Convert the expressions:
  $$8200\left(\frac{49}{50}\right)^{9}$$
- step3: Simplify:
  $$8200\times \frac{49^{9}}{50^{9}}$$
- step4: Rewrite the expression:
  $$50\times 164\times \frac{49^{9}}{50^{9}}$$
- step5: Reduce the numbers:
  $$164\times \frac{49^{9}}{50^{8}}$$
- step6: Rewrite the expression:
  $$4\times 41\times \frac{49^{9}}{50^{8}}$$
- step7: Rewrite the expression:
  $$4\times 41\times \frac{49^{9}}{2^{8}\times 25^{8}}$$
- step8: Rewrite the expression:
  $$2^{2}\times 41\times \frac{49^{9}}{2^{8}\times 25^{8}}$$
- step9: Reduce the numbers:
  $$41\times \frac{49^{9}}{2^{6}\times 25^{8}}$$
- step10: Multiply the fractions:
  $$\frac{41\times 49^{9}}{64\times 25^{8}}$$
The value of the investment after 9 years is approximately $6836.73.
The investment will be worth approximately \$6,836.73 after 9 years.

Example 6: An investment of \( \$ 8200 \) loses value at a rate of \( 2 \% \) per year. Use an exponential function to find the value of the investment after 9 years.

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