To determine the annual rate of change for Bruce's investment, we can use the formula for compound interest growth. Let's break down the problem step by step.
1. Identify the initial and final values of the investment:
- Initial value (Principal) [tex]\( P = \$ 24,000 \)[/tex]
- Final value [tex]\( A = \$ 108,595 \)[/tex]
- Number of years [tex]\( t = 20 \)[/tex]
2. Set up the formula for compound interest:
The general formula for the future value of an investment is given by:
[tex]\[
A = P \times (1 + r)^t
\][/tex]
where:
- [tex]\( A \)[/tex] is the final amount
- [tex]\( P \)[/tex] is the principal amount
- [tex]\( r \)[/tex] is the annual rate of change (expressed as a decimal)
- [tex]\( t \)[/tex] is the number of years
3. Rearrange the formula to solve for the annual rate [tex]\( r \)[/tex]:
[tex]\[
\frac{A}{P} = (1 + r)^t
\][/tex]
Take the t-th root of both sides to isolate [tex]\( 1 + r \)[/tex]:
[tex]\[
\left( \frac{A}{P} \right)^{\frac{1}{t}} = 1 + r
\][/tex]
Solve for [tex]\( r \)[/tex]:
[tex]\[
r = \left( \frac{A}{P} \right)^{\frac{1}{t}} - 1
\][/tex]
4. Substitute the known values into the equation:
[tex]\[
r = \left( \frac{108,595}{24,000} \right)^{\frac{1}{20}} - 1
\][/tex]
5. Calculate the result:
[tex]\[
r \approx \left( 4.5248 \right)^{\frac{1}{20}} - 1
\][/tex]
[tex]\[
r \approx 0.07840012466618695
\][/tex]
6. Convert the annual rate to a percentage:
[tex]\[
\text{Annual rate} = r \times 100 \approx 7.840012466618695 \%
\][/tex]
7. Round to two decimal places:
[tex]\[
\text{Annual rate} \approx 7.84\%
\][/tex]
Therefore, the annual rate of change was [tex]\( \boxed{7.84\%} \)[/tex].
