Question
Shota invests
\[\$2000\] in a certificate of deposit that earns
\[2\%\] in interest each year.
Write a function that gives the total value
\[V(t)\], in dollars, of the investment
\[t\] years from now.
Do not enter commas in your answer.
\[V(t)=\]
\[\$2000\] in a certificate of deposit that earns
\[2\%\] in interest each year.
Write a function that gives the total value
\[V(t)\], in dollars, of the investment
\[t\] years from now.
Do not enter commas in your answer.
\[V(t)=\]
Answers
GPT-4o mini
To calculate the total value \( V(t) \) of Shota's investment in a certificate of deposit that earns \( 2\% \) interest each year, we can use the formula for compound interest. The formula is given by:
\[
V(t) = P(1 + r)^t
\]
where:
- \( V(t) \) is the total value after \( t \) years,
- \( P \) is the principal amount (initial investment),
- \( r \) is the annual interest rate (as a decimal),
- \( t \) is the time in years.
In this case:
- \( P = 2000 \)
- \( r = 0.02 \) (which is \( 2\% \) expressed as a decimal)
Plugging these values into the formula gives us:
\[
V(t) = 2000(1 + 0.02)^t
\]
Simplifying further, we have:
\[
V(t) = 2000(1.02)^t
\]
So, the function that gives the total value \( V(t) \) of the investment \( t \) years from now is:
\[
V(t) = 2000(1.02)^t
\]
\[
V(t) = P(1 + r)^t
\]
where:
- \( V(t) \) is the total value after \( t \) years,
- \( P \) is the principal amount (initial investment),
- \( r \) is the annual interest rate (as a decimal),
- \( t \) is the time in years.
In this case:
- \( P = 2000 \)
- \( r = 0.02 \) (which is \( 2\% \) expressed as a decimal)
Plugging these values into the formula gives us:
\[
V(t) = 2000(1 + 0.02)^t
\]
Simplifying further, we have:
\[
V(t) = 2000(1.02)^t
\]
So, the function that gives the total value \( V(t) \) of the investment \( t \) years from now is:
\[
V(t) = 2000(1.02)^t
\]