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.

Result

x =  88.7 m

#### Solution:

$V=3500000 \ \\ V=\pi h/6 \cdot \ ( 3r^2+h^2) \ \\ 6V/\pi=h ( 3r^2+h^2) \ \\ h^3+ 67500 \ h - (6V/\pi)=0 \ \\ h^3+67500h-6684507.60899=0 \ \\ \ \\ h_{1}=88.6933426414 \doteq 88.6933 \ \\ h_{2}=-44.3466713+270.9241255i \ \\ h_{3}=-44.3466713-270.9241255i \ \\ \ \\ h>0 \ \\ \ \\ x=h_{1}=88.6933 \doteq 88.6933 \doteq 88.7 \ \text{m}$

Equation is non-linear.
$h^3+67500h-6684507.60899=0$
h1 = 88.6933426414
h2 = -44.3466713+270.9241255i
h3 = -44.3466713-270.9241255i

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