Question

Create a formula for the amount of money in Nico’s savings account over time. (Show all your work)

When Nico was born, his grandparents put $3,500 in a special savings account. The amount of money in the savings account increases by 4% APY (annual percentage yield). Nico tracks how much money is in the account over time.

Answer 1

all in one paragraph

Answer 2

To create a formula for the amount of money in Nico's savings account over time, we will use the formula for compound interest. The formula for the future value \( A \) of an investment after \( t \) years with compound interest is given by:

\[
A = P(1 + r)^t
\]

where:
- \( A \) is the amount of money accumulated after \( n \) years, including interest.
- \( P \) is the principal amount (the initial amount of money).
- \( r \) is the annual interest rate (decimal).
- \( t \) is the time the money is invested or borrowed for, in years.

**Step 1: Identify the values for \( P \), \( r \), and \( t \)**

From the problem:
- The principal amount \( P = 3500 \)
- The annual interest rate \( r = 4\% = 0.04 \)
- The time \( t \) is expressed in years, which is a variable in our formula.

**Step 2: Substitute the values into the formula**

Substituting the values for \( P \) and \( r \) into the compound interest formula gives us:

\[
A = 3500(1 + 0.04)^t
\]

**Step 3: Simplify the formula**

Now we can simplify the expression a bit:

\[
A = 3500(1.04)^t
\]

This formula \( A = 3500(1.04)^t \) represents the amount of money in Nico’s savings account after \( t \) years.

**Summary of the Final Formula:**

The amount of money in Nico's savings account over time is given by:

\[
A(t) = 3500(1.04)^t
\]

Where:
- \( A(t) \) is the amount in the savings account after \( t \) years.
- \( t \) is the number of years since Nico was born.

Answer 3

To create a formula for the amount of money in Nico's savings account over time, we use the compound interest formula \( A = P(1 + r)^t \), where \( A \) is the amount of money accumulated after \( t \) years, \( P \) is the principal amount (initial deposit), \( r \) is the annual interest rate (as a decimal), and \( t \) is the time in years. In this case, Nico's grandparents deposited $3,500 into the account, with an annual percentage yield (APY) of 4%, or 0.04 in decimal form, which gives us the principal \( P = 3500 \) and the rate \( r = 0.04 \). Substituting these values into the formula results in \( A = 3500(1 + 0.04)^t \). Simplifying this yields \( A = 3500(1.04)^t \), which represents the amount of money in Nico’s savings account after \( t \) years.