39.375*10^6=210/0.208*x^3

asked by guest
on Oct 20, 2024 at 7:16 pm



You asked:

Solve the equation \(39.375 \cdot {10}^{6} = \frac{210}{0.208} \cdot {x}^{3}\) for the variable \(x\).

MathBot Answer:

The solutions are given by the formula: \[x = 10 \cdot \sqrt[3]{39} \cos{\left(\frac{2 \pi k}{3} \right)} + 10 \cdot \sqrt[3]{39} \sin{\left(\frac{2 \pi k}{3} \right)} i\] 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: \[\begin{aligned}x &= 10 \cdot \sqrt[3]{39} \\&\approx 33.912114\end{aligned}\]

Solutions with nonzero imaginary part: \[\begin{aligned}x &= - 5 \cdot \sqrt[3]{39} + 5 \cdot \sqrt[3]{13} \cdot 3^{\frac{5}{6}} i \\&\approx -16.956057 + 29.368753 i\\x &= - 5 \cdot \sqrt[3]{39} - 5 \cdot \sqrt[3]{13} \cdot 3^{\frac{5}{6}} i \\&\approx -16.956057 -29.368753 i\end{aligned}\]

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

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