Suppose you deposit $5,000 in a savings account that earns 3% annual interest. If you make no other withdrawals or deposits, how many years will it take the account balance to reach at least $6,000? (1 point) Responses 10 years 10 years 6 years 6 years 7 years 7 years 4 years
To find out how many years it will take the account balance to reach at least $6,000, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the final amount,
P = the principal amount (initial deposit),
r = annual interest rate (in decimal form),
n = number of times interest is compounded per year,
t = number of years.
We have:
P = $5,000
r = 3% or 0.03
A = $6,000
To determine the number of years, we can plug in these values and solve for t:
$6,000 = $5,000(1 + 0.03/n)^(n*t)
Rearranging the equation, we get:
1.2 = (1 + 0.03/n)^(n*t)
To find a solution for t, we can try different values of n as it represents the number of times interest is compounded per year. Let's start with n = 1 (compounded annually):
1.2 = (1 + 0.03)^(1*t)
1.2 = (1.03)^t
Taking the natural logarithm (ln) of both sides:
ln(1.2) = ln(1.03^t)
ln(1.2) = t * ln(1.03)
Now, we calculate t:
t = ln(1.2) / ln(1.03)
t ≈ 9.402
Therefore, it will take approximately 9.4 years for the account balance to reach at least $6,000. Rounded to the nearest whole number, the correct response is 9 years.