π(h/2)² * h + (2/3)π(h/2)³ = 880/21

asked by guest
on Oct 23, 2024 at 7:28 am



You asked:

Solve the equation \(\pi \cdot {\left( \frac{h}{2} \right)}^{2} \cdot h + \frac{2}{3} \cdot \pi \cdot {\left( \frac{h}{2} \right)}^{3} = \frac{880}{21}\) for the variable \(h\).

MathBot Answer:

The solutions are given by the formula: \[h = \frac{2 \cdot 7^{\frac{2}{3}} \cdot \sqrt[3]{110} \cos{\left(\frac{2 \pi k}{3} \right)}}{7 \sqrt[3]{\pi}} + \frac{2 \cdot 7^{\frac{2}{3}} \cdot \sqrt[3]{110} \sin{\left(\frac{2 \pi k}{3} \right)}}{7 \sqrt[3]{\pi}} i\] 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: \[\begin{aligned}h &= \frac{2 \cdot 7^{\frac{2}{3}} \cdot \sqrt[3]{110}}{7 \sqrt[3]{\pi}} \\&\approx 3.4204107\end{aligned}\]

Solutions with nonzero imaginary part: \[\begin{aligned}h &= - \frac{\sqrt[3]{110} \cdot 7^{\frac{2}{3}}}{7 \sqrt[3]{\pi}} + \frac{\sqrt[3]{110} \sqrt{3} \cdot 7^{\frac{2}{3}} i}{7 \sqrt[3]{\pi}} \\&\approx -1.7102053 + 2.9621625 i\\h &= - \frac{\sqrt[3]{110} \cdot 7^{\frac{2}{3}}}{7 \sqrt[3]{\pi}} - \frac{\sqrt[3]{110} \sqrt{3} \cdot 7^{\frac{2}{3}} i}{7 \sqrt[3]{\pi}} \\&\approx -1.7102053 -2.9621625 i\end{aligned}\]

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

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