Stadium

A domed stadium is in the shape of spherical segment with a base radius of 150 m. The dome must contain a volume of 3500000 m³. Determine the height of the dome at its centre to the nearest tenth of a meter.

Correct result:

x =  88.7 m

Solution:

$$\begin{gathered} V=3500000 \\ V=\pi h\mathrm{/}6\cdot (3r^2+h^2) \\ 6V\mathrm{/}\pi =h(3r^2+h^2) \\ {h}^3+67500h-(6V\mathrm{/}\pi )=0 \\ {h}^3+67500h-6684507.60899=0 \\ {h}_1=88.6933426414\doteq 88.6933 \\ {h}_2=-44.3466713+270.9241255i \\ {h}_3=-44.3466713-270.9241255i \\ h>0 \\ x=h_1=88.6933=88.7\text{ m} \end{gathered}$$

Equation is non-linear.

Equation is not quadratic.

$h^3+67500h-6684507.60899=0$

h₁ = 88.693342641435
h₂ = -44.3466713+270.9241255i
h₃ = -44.3466713-270.9241255i

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