solvefor r,x= 4/3 pir^3

Lösung

löse nach

r, x = \frac{4}{3} π r^{3}

r = 3 \frac{3 x}{4 π}, r = - \frac{4 ^{\frac{2}{3}} 3 3 π ^{\frac{2}{3}} 3 x}{8 π} + i \frac{3 ^{\frac{5}{6}} \cdot 4 ^{\frac{2}{3}} π ^{\frac{2}{3}} 3 x}{8 π}, r = - \frac{4 ^{\frac{2}{3}} 3 3 π ^{\frac{2}{3}} 3 x}{8 π} - i \frac{3 ^{\frac{5}{6}} \cdot 4 ^{\frac{2}{3}} π ^{\frac{2}{3}} 3 x}{8 π}

Schritte zur Lösung

x = \frac{4}{3} π r^{3}

Tausche die Seiten

\frac{4}{3} π r^{3} = x

Vereinfache

\frac{4}{3} π : \frac{4 π}{3}

\frac{4 π}{3} r^{3} = x

Multipliziere beide Seiten mit

3

4 π r^{3} = 3 x

Teile beide Seiten durch

4 π

r^{3} = \frac{3 x}{4 π}

Für

x^{3} = f (a)

sind die Lösungen

x = 3 f (a), 3 f (a) \frac{- 1 - 3 i}{2}, 3 f (a) \frac{- 1 + 3 i}{2}

r = 3 \frac{3 x}{4 π}, r = 3 \frac{3 x}{4 π} \frac{- 1 + 3 i}{2}, r = 3 \frac{3 x}{4 π} \frac{- 1 - 3 i}{2}

Vereinfache

3 \frac{3 x}{4 π} \frac{- 1 + 3 i}{2} : - \frac{4 ^{\frac{2}{3}} 3 3 π ^{\frac{2}{3}} 3 x}{8 π} + i \frac{3 ^{\frac{5}{6}} \cdot 4 ^{\frac{2}{3}} π ^{\frac{2}{3}} 3 x}{8 π}

Vereinfache

3 \frac{3 x}{4 π} \frac{- 1 - 3 i}{2} : - \frac{4 ^{\frac{2}{3}} 3 3 π ^{\frac{2}{3}} 3 x}{8 π} - i \frac{3 ^{\frac{5}{6}} \cdot 4 ^{\frac{2}{3}} π ^{\frac{2}{3}} 3 x}{8 π}

r = 3 \frac{3 x}{4 π}, r = - \frac{4 ^{\frac{2}{3}} 3 3 π ^{\frac{2}{3}} 3 x}{8 π} + i \frac{3 ^{\frac{5}{6}} \cdot 4 ^{\frac{2}{3}} π ^{\frac{2}{3}} 3 x}{8 π}, r = - \frac{4 ^{\frac{2}{3}} 3 3 π ^{\frac{2}{3}} 3 x}{8 π} - i \frac{3 ^{\frac{5}{6}} \cdot 4 ^{\frac{2}{3}} π ^{\frac{2}{3}} 3 x}{8 π}

x^{3} + 3 x^{2} - 3 x - 9 = 0

(x + 1) (x - 2) (x + 3) = 0

x^{2} (x + 1) - 64 (x + 1) = 0

x^{3} + 6 x^{2} = 16 x

x^{3} = 100