After 36 months (3 years), how much money will be in the account?
Understand the Problem
The question is asking to analyze a bank account situation where the initial amount triples every month. We need to create a table, graph the growth, and formulate an equation to represent the situation. Finally, we are asked to determine the amount in the account after 36 months.
Answer
After 36 months, the amount will be approximately $7.62559748 \times 10^{17}$.
Answer for screen readers
After 36 months, the amount in the account will be approximately:
$$ 7.62559748 \times 10^{17} \text{ dollars} $$
Steps to Solve
- Create a Table
To begin, we can calculate the amount in the bank account for the first few months. The amount triples each month.
| Month | Amount (in $) |
|---|---|
| 0 | $10 |
| 1 | $30 |
| 2 | $90 |
| 3 | $270 |
| 4 | $810 |
| 5 | $2430 |
| ... | ... |
| 36 | ? |
To calculate subsequent months, use the formula:
$$ \text{Amount}{n} = 3 \times \text{Amount}{n-1} $$
- Continue Table Calculation
Continue calculating the values for all months up to 36:
- Month 0: $10
- Month 1: $10 × 3 = $30
- Month 2: $30 × 3 = $90
- Month 3: $90 × 3 = $270
- Month 4: $270 × 3 = $810
- Month 5: $810 × 3 = $2430
- Continue until Month 36.
- Formulate an Equation
The amount in the account can be described with the exponential function:
$$ A(n) = 10 \times 3^n $$
Where $A(n)$ is the amount after $n$ months.
- Calculate Amount After 36 Months
Now, plugging in $n = 36$ in our equation:
$$ A(36) = 10 \times 3^{36} $$
Calculating this gives the amount after 36 months.
- Final Calculation
Now we will calculate $3^{36}$ using a calculator:
$$ A(36) = 10 \times 3^{36} \approx 10 \times 7.62559748 \times 10^{16} $$
After 36 months, the amount in the account will be approximately:
$$ 7.62559748 \times 10^{17} \text{ dollars} $$
More Information
This situation illustrates the power of exponential growth, where the initial amount grows rapidly due to being multiplied consistently. Starting with a mere $10 leads to an astounding amount due to the tripling effect.
Tips
- Forgetting to multiply by the previous amount in each step.
- Miscalculating the exponential growth by confusing powers of three.
- Not representing the growth formula correctly.
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