Questions: Suppose you have 1100 deposited at 2.55% compounded quarterly. About how long will it take your balance to increase to 2800? years Round your answer to the nearest tenth of a year

Suppose you have 1100 deposited at 2.55% compounded quarterly. About how long will it take your balance to increase to 2800?
years
Round your answer to the nearest tenth of a year

Transcript text: Suppose you have $\$ 1100$ deposited at $2.55 \%$ compounded quarterly. About how long will it take your balance to increase to $\$ 2800$ ? $\square$ years Round your answer to the nearest tenth of a year

Solution

Solution Steps

Step 1: Understand the Problem

We need to find the time required for an initial deposit $P$ to grow to a future value $A$ with an annual interest rate $r$ compounded $n$ times per year.

Step 2: Apply the Future Value Formula

The future value $A$ is calculated using the formula: $A = P(1 + \frac{r}{n})^{nt}$.

Step 3: Rearrange the Formula to Solve for $t$

To find $t$, we rearrange the formula to: $t = \frac{\log(A/P)}{n \log(1 + r/n)}$.

Step 4: Substitute the Values and Calculate

Substituting the given values: $P = 1100$, $A = 2800$, $r = 0.0255$, and $n = 4$, we calculate $t$.

Step 5: Calculation

Using the formula: $t = \frac{\log(2800/1100)}{4 \log(1 + 0.0255/4)}$, we find $t = 36.8$ years.

Final Answer:

The time required for the initial deposit to grow to the future value is approximately 36.8 years.

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