Question

7700 dollars is placed in an account with an annual interest rate of 8.25%. To the nearest tenth of a year, how long will it take for the account value to reach 44700 dollars? Answer: Submit Answer attempt 1 out of 5

Answer 1

It will take approximately 17.3 years for the account value to reach $44,700.

Step-by-step explanation:

To calculate the time required for the account value to reach $44,700, the compound interest formula is employed:
`\[A = P \left(1 + (r)/(n)\right)^(nt),\]` where (A) is the future value, (P) is the principal amount, (r) is the annual interest rate, (n) is the number of times interest is compounded per year, and (t) is the time in years. Given an initial deposit ((P)) of $7,700, a future value ((A)) of $44,700, an annual interest rate ((r)) of 8.25% (or 0.0825 as a decimal), and interest compounded annually ((n = 1)), the goal is to solve for (t).

By rearranging the formula to solve for (t), the computation yields approximately 17.3 years. This implies that, at an annual interest rate of 8.25%, it will take about 17.3 years for the initial deposit of $7,700 to grow to $44,700.

Understanding compound interest is vital in financial planning, offering insights into how investments accumulate over time. This calculation illustrates the impact of interest rates on the growth of an investment, aiding individuals in making informed decisions about savings and long-term financial goals.

Answer 2

To determine the time required for an investment to reach a specific value with compound interest, the compound interest formula is used. It will take approximately 21.7 years for an initial investment of $7,700 to grow to $44,700 at an annual interest rate of 8.25%, compounded annually.

Step-by-step explanation:

The question is asking how long it takes for an initial investment to grow to a specified amount given a certain annual interest rate. This is a problem that involves compound interest and can be solved using the formula for compound growth, which is A = P(1 + r/n)^(nt), where A is the final amount, P is the initial principal balance, r is the interest rate, t is the time the money is invested for, and n is the number of times that interest is compounded per year.

In this case, the money is compounded annually, so n = 1. We are given the initial amount (P = $7,700), the final amount (A = $44,700), and the annual interest rate (r = 8.25% or 0.0825 in decimal form). We need to find the time, t.

Let's rearrange the formula to solve for t:

44700 = 7700(1 + 0.0825)^t

Now, divide both sides by 7700:

5.8 = (1 + 0.0825)^t

To find t, we take the logarithm of both sides:

ln(5.8) = t * ln(1.0825)

Now divide both sides by ln(1.0825) to solve for t:

t = ln(5.8) / ln(1.0825)

Using a calculator, we find that t ≈ 21.7

Therefore, it will take approximately 21.7 years for the account value to reach $44,700, to the nearest tenth of a year.