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]
### 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]