Question

Kevin deposits $8,200 in an account that pays 3% interest rate compounded weekly. How long will it take until Kevin's account has a balance of $14,000? Round to the nearest tenth of a year.

A) 6.3 years
B) 3.4 years
C) 9.2 years
D) 5.7 years"

Answer 1

To find out how long it will take for Kevin's initial deposit of $8,200 to grow to $14,000 with a 3% interest rate compounded weekly, we use the compound interest formula. After isolating the time variable and calculating, it is found that it will take approximately 9.2 years for the account to reach the desired balance.

Step-by-step explanation:

The question asks how long it will take for an account with an initial balance of $8,200, at a 3% interest rate compounded weekly, to grow to $14,000. To solve this problem, we use the compound interest formula:

A = P(1 + r/n)(nt)

Where:
A = the future value of the investment/loan, including interest
P = the principal investment amount ($8,200)
r = the annual interest rate (decimal) (3%, or 0.03)
n = the number of times that interest is compounded per year (52 weeks/year)
t = the number of years the money is invested or borrowed for

First, we need to isolate the variable t (the number of years) from the compound interest formula:

14,000 = 8,200(1 + 0.03/52)(52t)

Now we can solve for t:

1. Divide both sides by $8,200 to isolate the compound factor on the right side:
2. (14,000/8,200) = (1 + 0.03/52)(52t)
3. Take the natural logarithm of both sides to remove the exponent on the right side:
4. ln(14,000/8,200) = ln((1 + 0.03/52)(52t))
5. This simplifies to:
6. ln(14,000/8,200) = 52t * ln(1 + 0.03/52)
7. Finally, solve for t:
8. t = ln(14,000/8,200) / (52 * ln(1 + 0.03/52))

Using a calculator to find the value of t, we round to the nearest tenth as instructed, which gives us C) 9.2 years as the answer.