5=(1+r/100)^3

asked by guest
on Jan 01, 2025 at 1:54 am



You asked:

Solve the equation \(5 = {\left( 1 + \frac{r}{100} \right)}^{3}\) for the variable \(r\).

MathBot Answer:

The solutions are given by the formula: \[r = \left(-100 + 100 \cdot \sqrt[3]{5} \cos{\left(\frac{2 \pi k}{3} \right)}\right) + 100 \cdot \sqrt[3]{5} \sin{\left(\frac{2 \pi k}{3} \right)} i\] where \(k\) is an integer in \(0 \le k < 3\).

There is \(1\) real solution.

There are \(2\) solutions with nonzero imaginary part.

Real solutions: \[\begin{aligned}r &= -100 + 100 \cdot \sqrt[3]{5} \\&\approx 70.997595\end{aligned}\]

Solutions with nonzero imaginary part: \[\begin{aligned}r &= -100 - 50 \cdot \sqrt[3]{5} + 50 \sqrt{3} \cdot \sqrt[3]{5} i \\&\approx -185.4988 + 148.08826 i\\r &= -100 - 50 \cdot \sqrt[3]{5} - 50 \sqrt{3} \cdot \sqrt[3]{5} i \\&\approx -185.4988 -148.08826 i\end{aligned}\]

\(i\) is the imaginary unit, defined as \(i^2 = -1\).

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