Final answer:
To accumulate $13,000 in three years in an account that earns 8% compounded semiannually, approximately $9,684.37 should be deposited today.
Step-by-step explanation:
To determine how much money should be deposited today in an account that earns 8% compounded semiannually to accumulate to $13,000 in three years, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = Future value
P = Principal (initial deposit)
r = Interest rate (as a decimal)
n = Number of times interest is compounded per year
t = Number of years
In this case, we have:
A = $13,000
P = ?
r = 8% or 0.08
n = 2 (compounded semiannually means twice a year)
t = 3 years
Substituting these values into the formula, we get:
$13,000 = P(1 + 0.08/2)^(2*3)
First, simplify the exponent:
$13,000 = P(1 + 0.04)^6
Then, distribute the exponent:
$13,000 = P(1.04)^6
Next, divide both sides by (1.04)^6 to solve for P:
P = $13,000 / (1.04)^6
Using a calculator, we find:
P ≈ $9,684.37
Therefore, approximately $9,684.37 should be deposited today in the account to accumulate to $13,000 in three years.