Explanation:
To determine how long it will take for your initial deposit of $5,700 to grow to $6,900 at an annual interest rate of 5.4% compounded daily, you can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A is the future amount ($6,900 in this case).
P is the principal amount ($5,700).
r is the annual interest rate (5.4% or 0.054 as a decimal).
n is the number of times the interest is compounded per year (daily, so n = 365).
t is the number of years we want to find.
Now, let's plug in the values and solve for t:
6,900 = 5,700(1 + 0.054/365)^(365t)
First, let's simplify the fraction inside the parentheses:
6,900 = 5,700(1 + 0.00014794521)^(365t)
Now, calculate the expression inside the parentheses:
1 + 0.00014794521 = 1.00014794521
6,900 = 5,700(1.00014794521)^(365t)
Next, divide both sides by 5,700:
6,900 / 5,700 = (1.00014794521)^(365t)
1.21052631579 = (1.00014794521)^(365t)
Now, take the natural logarithm (ln) of both sides to solve for t:
ln(1.21052631579) = ln((1.00014794521)^(365t))
Using the property of logarithms, we can bring down the exponent:
ln(1.21052631579) = 365t * ln(1.00014794521)
Now, divide by 365ln(1.00014794521) to isolate t:
t = ln(1.21052631579) / (365 * ln(1.00014794521))
Now, calculate the right side of the equation:
t ≈ 6.8257 years
So, it will take approximately 6.8257 years for your balance to increase to $6,900 when compounded daily at an annual interest rate of 5.4%.