478.29= 3.14*D^2/4*3D

asked by guest
on Dec 30, 2024 at 2:34 pm



You asked:

Solve the equation \(478.29 = \frac{3.14 {D}^{2}}{4} \cdot 3 D\) for the variable \(D\).

MathBot Answer:

The solutions are given by the formula: \[D = \frac{157^{\frac{2}{3}} \cdot \sqrt[3]{31886} \cos{\left(\frac{2 \pi k}{3} \right)}}{157} + \frac{157^{\frac{2}{3}} \cdot \sqrt[3]{31886} \sin{\left(\frac{2 \pi k}{3} \right)}}{157} i\] 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: \[\begin{aligned}D &= \frac{157^{\frac{2}{3}} \cdot \sqrt[3]{31886}}{157} \\&\approx 5.8780525\end{aligned}\]


Solutions with nonzero imaginary part: \[\begin{aligned}D &= - \frac{157^{\frac{2}{3}} \cdot \sqrt[3]{31886}}{314} + \frac{157^{\frac{2}{3}} \sqrt{3} \cdot \sqrt[3]{31886} i}{314} \\&\approx -2.9390263 + 5.0905428 i\\D &= - \frac{157^{\frac{2}{3}} \cdot \sqrt[3]{31886}}{314} - \frac{157^{\frac{2}{3}} \sqrt{3} \cdot \sqrt[3]{31886} i}{314} \\&\approx -2.9390263 -5.0905428 i\end{aligned}\]


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