Certainly! Let's break down the problem step-by-step and solve it.
### Part A
A. What are the principal and the interest rate?
- Principal amount, [tex]\( P \)[/tex], is the initial amount of money invested.
- Interest rate, [tex]\( r \)[/tex], is the annual rate at which the investment grows.
Looking at the problem:
- The principal amount [tex]\( P \)[/tex] is [tex]$25.
- The annual interest rate \( r \) is 4%, which is written as a decimal \( 0.04 \).
### Part B
B. Write a function that represents Tobie's account balance after \( t \) years.
The formula for the compound interest when interest is compounded annually is given by:
\[ A(t) = P(1 + r)^t \]
Where:
- \( A(t) \) is the amount of money in the account after \( t \) years.
- \( P \) is the principal amount.
- \( r \) is the annual interest rate.
- \( t \) is the number of years the money is invested.
So, substituting the values of \( P \) and \( r \):
\[ A(t) = 25(1 + 0.04)^t \]
### Part C
C. Complete the table for the given values of \( t \). Round \( A(t) \) values to the nearest integer.
Now we calculate \( A(t) \) for \( t = 0, 5, \) and \( 10 \):
1. When \( t = 0 \):
\[ A(0) = 25(1 + 0.04)^0 = 25 \]
(Any number raised to the power of 0 is 1, so 25 * 1 = 25)
2. When \( t = 5 \):
\[ A(5) = 25(1 + 0.04)^5 \]
Using the given result:
\[ A(5) \approx 30 \]
3. When \( t = 10 \):
\[ A(10) = 25(1 + 0.04)^10 \]
Using the given result:
\[ A(10) \approx 37 \]
Thus, the completed table is:
\[
\begin{tabular}{|c|c|}
\hline
$[/tex]t[tex]$ & $[/tex]A(t)[tex]$ \\
\hline
0 & 25 \\
\hline
5 & 30 \\
\hline
10 & 37 \\
\hline
\end{tabular}
\]
In summary:
- The principal amount \( P \) is $[/tex]25.
- The annual interest rate [tex]\( r \)[/tex] is 0.04 (or 4%).
- The function representing the account balance after [tex]\( t \)[/tex] years is [tex]\( A(t) = 25(1 + 0.04)^t \)[/tex].
- The table with [tex]\( A(t) \)[/tex] values for [tex]\( t = 0, 5, \)[/tex] and [tex]\( 10 \)[/tex] is:
[tex]\[
\begin{tabular}{|c|c|}
\hline
$t$ & $A(t)$ \\
\hline
0 & 25 \\
\hline
5 & 30 \\
\hline
10 & 37 \\
\hline
\end{tabular}
\][/tex]
