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\begin{tabular}{|c|c|}
\hline Time (years) & Total amount (dollars) \\
\hline 0 & 604.00 \\
\hline 1 & 606.42 \\
\hline 2 & 608.84 \\
\hline
\end{tabular}

Rosa opened a savings account at a bank. The table shows the exponential relationship between the time [tex]$t$[/tex], in years, since Rosa opened the account and the total amount [tex]$n$[/tex], in dollars, in the account. If Rosa made no additional deposits or withdrawals, which of the following equations best represents the relationship between [tex]$t$[/tex] and [tex]$n$[/tex]?

A. [tex]n=(1+604)^t[/tex]

B. [tex]n=(1+0.004)^t[/tex]

C. [tex]n=604(1+0.004)^t[/tex]

D. [tex]n=0.004(1+604)^t[/tex]



Answer :

GinnyAnswer
Let's break down the steps to determine which equation best represents the relationship between the time [tex]$t$[/tex] (in years) and the total amount [tex]$n$[/tex] (in dollars):

1. Identify the Initial Amount:
- From the table, the amount at time [tex]$t = 0$[/tex] is \[tex]$604.00. Therefore, this is the initial amount $[/tex]n_0[tex]$. 2. Calculate the Growth Rate: - In one year, the amount grows from \$[/tex]604.00 to \[tex]$606.42. - Growth rate per year is given by the formula: \[ \text{Growth Rate} = \frac{\text{Amount at time } t = 1}{\text{Initial Amount}} - 1 \] - Plugging in the values: \[ \text{Growth Rate} = \frac{606.42}{604.00} - 1 = 0.0040066225165562575 \] 3. Equation Formulation: - The general form of the exponential growth equation is: \[ n = n_0 \cdot (1 + r)^t \] - Where: - \( n_0 \) is the initial amount (\$[/tex]604.00)
- [tex]\( r \)[/tex] is the growth rate (0.0040066225165562575)
- [tex]\( t \)[/tex] is time in years

- Hence, substituting the values:
[tex]\[ n = 604.00 \cdot (1 + 0.0040066225165562575)^t \][/tex]

4. Verification with Given Data:
- For [tex]$t = 1$[/tex] and [tex]$t = 2$[/tex]:
- When [tex]$t = 1$[/tex]:
[tex]\[ n = 604 \cdot (1 + 0.0040066225165562575)^1 = 606.42 \quad \text{(approximately)} \][/tex]
- When [tex]$t = 2$[/tex]:
[tex]\[ n = 604 \cdot (1 + 0.0040066225165562575)^2 = 608.8496960264899 \quad \text{(approximately 608.84)} \][/tex]

5. Compare with Given Options:
- Now we check which provided option matches the formulated equation:
[tex]\[ n = 604(1 + 0.004)^t \][/tex]

- Clearly, the best fitting option is (C):
[tex]\[ \boxed{n = 604(1 + 0.004)^t} \][/tex]