Final answer:
To find the time required for an investment to grow to a certain amount at a given interest rate, we can use the formula for compound interest. In this case, the investment starts with $5,000 and grows to $6,900 at an interest rate of 7.5% p.a., compounded semiannually. The time required for the investment to grow to $6,900 is approximately 11 years.
Step-by-step explanation:
To find the time required for an investment to grow to a certain amount, we can use the formula for compound interest: A = P(1 + r/n)^(nt), where A is the final amount, P is the principal amount, r is the interest rate, n is the number of times interest is compounded per year, and t is the number of years. In this case, we know that the principal is $5,000, the interest rate is 7.5%, and the final amount is $6,900.
The interest is compounded semiannually, so n = 2. Substituting the values into the formula, we have:
$6,900 = $5,000(1 + 0.075/2)^(2t)
Divide both sides of the equation by $5,000:
1.38 = (1 + 0.0375)^2t
Take the natural logarithm of both sides of the equation:
ln(1.38) = 2t * ln(1 + 0.0375)
Divide both sides of the equation by 2 * ln(1 + 0.0375):
t = ln(1.38) / (2 * ln(1.0375))
Using a calculator, we find that t is approximately 10.76 years. Rounding to the nearest year, the time required for the investment to grow to $6,900 is 11 years.