Shota invests [tex]\$2000[/tex] in a certificate of deposit that earns [tex]2\%[/tex] in interest each year.

Write a function that gives the total value [tex]V(t)[/tex], in dollars, of the investment [tex]t[/tex] years from now. Do not enter commas in your answer.

[tex]V(t) = 2000(1 + 0.02)^t[/tex]



Answer :

Certainly! Let's analyze the problem and derive the formula for the total value of the investment over [tex]$t$[/tex] years.

### Step-by-Step Solution:

1. Initial Investment and Interest Rate:
- Initial Investment (P): \[tex]$2000 - Annual Interest Rate (r): 2%, which can be written as 0.02 in decimal form. 2. Compound Interest Formula: The formula for compound interest, where interest is compounded annually, can be expressed as: \[ V(t) = P \left(1 + r\right)^t \] In this case: - \( P = 2000 \) - \( r = 0.02 \) - \( t \) is the number of years 3. Substitute the Known Values: \[ V(t) = 2000 \left(1 + 0.02\right)^t \] 4. Simplify the Expression: \[ V(t) = 2000 \left(1.02\right)^t \] 5. Conclusion: The function that represents the total value of the investment $[/tex]V(t)[tex]$, in dollars, after $[/tex]t[tex]$ years is: \[ V(t) = 2000 \left(1.02\right)^t \] So, the final formula that gives the value of Shota's investment $[/tex]t$ years from now is:

[tex]\[ \boxed{V(t) = 2000 \left(1.02\right)^t} \][/tex]