Question
Writing an equation that models exponential growth or decay A principal of \( \$ 3600 \) was invested at \( 4.25 \% \) interest, compounded annually. Let \( t \) be the number of years since the start of the investment. Let \( y \) be the value of the investment, in dollars. Write an exponential function showing the relationship between \( y \) and \( t \). W
Ask by Hall Carroll. in the United States
Mar 12,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The investment's value after \( t \) years is given by the equation:
\[
y = 3600 \times (1.0425)^t
\]
Solution
1. The formula for compound interest is given by
\[
y = P \left(1 + r\right)^t,
\]
where:
- \( P \) is the principal (initial amount),
- \( r \) is the annual interest rate (in decimal),
- \( t \) is the time in years, and
- \( y \) is the final amount after \( t \) years.
2. Substitute the given values:
- \( P = 3600 \),
- \( r = 4.25\% = 0.0425 \).
The equation becomes:
\[
y = 3600 \left(1 + 0.0425\right)^t.
\]
3. Simplify inside the parentheses:
\[
1 + 0.0425 = 1.0425.
\]
Therefore, the final equation is:
\[
y = 3600 \cdot \left(1.0425\right)^t.
\]
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Simplify this solution Beyond the Answer
The equation that models the value of the investment can be written as \( y = 3600(1 + 0.0425)^t \) or \( y = 3600(1.0425)^t \). In this equation, \( 3600 \) is the initial principal, \( 0.0425 \) represents the interest rate (4.25%), and \( t \) is the number of years. This function indicates that the investment grows exponentially over time due to the compounded interest. If you're looking to better understand exponential relationships, think of it as a "snowball effect." The more time \( t \) you have, the more interest you'll earn, which in turn earns more interest in subsequent years. Just remember, compounding is your best friend for growing investments!
