To find the balance in Jacob's account at the end of six years, we can break down the problem into two parts: the first four years and the remaining two years.
1. First, let's calculate the balance at the end of the first four years. Jacob initially deposited $25,000 in the savings account at 10% interest compounded semiannually. This means that interest is accrued twice a year.
To find the balance after four years, we need to calculate the value of the initial deposit plus the interest earned over the four-year period. The formula to calculate compound interest is:
A = P(1 + r/n)^(n*t)
where:
A = the final amount
P = the principal amount (initial deposit)
r = annual interest rate (as a decimal)
n = number of times the interest is compounded per year
t = number of years
In this case, P = $25,000, r = 10% or 0.10, n = 2 (compounded semiannually), and t = 4.
Plugging in the values, we have:
A = $25,000 * (1 + 0.10/2)^(2*4)
Simplifying the exponents:
A = $25,000 * (1 + 0.05)^8
Calculating the value inside the parentheses:
A = $25,000 * (1.05)^8
Using a calculator, we find:
A ≈ $38,698.50 (rounded to the nearest cent)
2. Now, let's calculate the balance at the end of the remaining two years. At the beginning of year 4, Jacob deposits an additional $40,000 at 10% interest compounded semiannually.
Using the same formula as before with the new principal amount, we have:
P = $38,698.50 (the balance at the end of four years) + $40,000 (additional deposit)
A = $38,698.50 + $40,000 * (1 + 0.10/2)^(2*2)
Simplifying:
A = $38,698.50 + $40,000 * (1.05)^4
Using a calculator, we find:
A ≈ $95,998.58 (rounded to the nearest cent)
Therefore, at the end of six years, the balance in Jacob's account is approximately $95,998.58.