Final answer:
To find the interest rate, use the formula for compound interest and solve for the unknown variable. In this case, the interest rate on the $5000 investment compounded daily is approximately 5.96%.
Step-by-step explanation:
To find the interest rate, we can use the formula for compound interest: A = P(1 + r/n)^(nt), where A is the final amount, P is the initial principal, r is the interest rate, n is the number of times compounded per year, and t is the number of years. In this case, the initial amount invested is $5000, the final amount is $6756, and the time period is 7 years.
Since the investment is compounded daily, n = 365. Plugging the values into the formula, we have: $6756 = $5000(1 + r/365)^(365*7). We can solve this equation to find the interest rate, r.
- Divide both sides of the equation by $5000: $6756/$5000 = (1 + r/365)^(365*7)
- Take the natural logarithm of both sides: ln($6756/$5000) = ln[(1 + r/365)^(365*7)]
- Use the property of logarithms: ln(x^y) = yln(x)
- Apply the property to the right side of the equation: ln($6756/$5000) = 7ln(1 + r/365)
- Divide both sides of the equation by 7: ln($6756/$5000)/7 = ln(1 + r/365)
- Raise e (the natural logarithm base) to the power of both sides: e^(ln($6756/$5000)/7) = e^(ln(1 + r/365))
- Use the property of logarithms: e^ln(x) = x
- Simplify both sides of the equation: ($6756/$5000)^(1/7) = 1 + r/365
- Subtract 1 from both sides of the equation: ($6756/$5000)^(1/7) - 1 = r/365
- Multiply both sides of the equation by 365: 365[($6756/$5000)^(1/7) - 1] = r
- Simplify and calculate: r ≈ 0.0596, or 5.96%
Therefore, the interest rate on the $5000 investment compounded daily is approximately 5.96%.