To answer the first question, we need to use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the final amount (balance)
P = the initial principal (deposit)
r = annual interest rate (in decimal)
n = number of times the interest is compounded per year
t = time in years
We know the following information:
P = $7,500
r = 0.036 (3.6% expressed as a decimal)
n = 4 (compounded quarterly)
A = $9,750
We need to find t, so we can rearrange the formula:
t = (log(A/P))/(n * log(1 + r/n))
Plugging in the values, we get:
t = (log(9,750/7,500))/(4 * log(1 + 0.036/4))
Using a calculator, we can find the value of t to be approximately 3.7 years.
Therefore, the account balance will be $9,750 around 3.7 years after the money is deposited.
Now, let's move on to the second question.
To calculate the initial deposit required to reach a balance of $12,000 in exactly 4 years and 6 months, we can rearrange the formula for compound interest:
P = A / (1 + r/n)^(nt)
Given the following information:
A = $12,000
r = (unknown)
n = (unknown)
t = 4.5 years
We are looking to find P, so we can rearrange and solve the formula:
P = 12,000 / (1 + r/n)^(4.5n)
Since we have two unknowns (r and n), we'll need more information to determine the exact values.
If you provide additional information, such as the compounding frequency (n) or an interest rate (r), we can calculate the initial deposit required.