- Substitute the values \(p=800\), \(a=2700\), and \(t=3\) into the given formula \(r = (\frac{a}{p})^{\frac{1}{t}} - 1\).
- Simplify the ratio to \(\frac{2700}{800} = 3.375\) and compute the cube root to get \(1.5\).
- Subtract 1 to find the decimal interest rate \(r = 0.5\).
- Convert the decimal to a percentage by multiplying by 100, which yields \(\boxed{50\%}\).
Explanation
- Analysis of the Problem Data
To determine the interest rate \(r\), we use the provided formula \(r = \left(\frac{a}{p}\right)^{\frac{1}{t}} - 1\). We are given the following values:
- Principal investment \(p = 800\)
- Final amount \(a = 2700\)
- Time period \(t = 3\) years
Our objective is to calculate the value of \(r\) and identify its percentage equivalent.
- Calculation of the Interest Rate
Substitute the given values into the formula:
\[r = \left(\frac{2700}{800}\right)^{\frac{1}{3}} - 1\]
First, simplify the fraction inside the parentheses:
\[\frac{2700}{800} = \frac{27}{8} = 3.375\]
Now, compute the cube root of \(3.375\):
\[(3.375)^{\frac{1}{3}} = 1.5\]
Subtract 1 from the result to find \(r\):
\[r = 1.5 - 1 = 0.5\]
- Conclusion
The decimal value calculated for the interest rate is \(r = 0.5\). To express this as a percentage, multiply by 100:
\[0.5 \times 100 = 50\%\]
Comparing this result to the given options:
- \(0.5\%\)
- \(50\%\)
- \(1.5\%\)
- \(150\%\)
The calculated interest rate is \(50\%\). The final answer is \(\boxed{50\%}\).
Examples
In finance, calculating interest rates is essential for understanding how an investment grows over time. For example, if you deposit a certain amount in a high-yield savings account or invest in a project, the formula \(r = (\frac{a}{p})^{\frac{1}{t}} - 1\) allows you to determine the annual growth rate (CAGR). This helps investors compare different financial opportunities and make informed decisions about where to allocate their capital based on expected returns.