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.

Result

x =  88.7 m

Solution:

V=3500000 V=πh/6⋅ (3r2+h2) 6V/π=h(3r2+h2) h3+67500 h−(6V/π)=0 h3+67500h−6684507.60899=0  h1=88.6933426414≐88.6933 h2=−44.3466713+270.9241255i h3=−44.3466713−270.9241255i  h>0  x=h1=88.6933≐88.6933≐88.7 mV=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.
Equation is not quadratic.
h3+67500h−6684507.60899=0h^3+67500h-6684507.60899=0
h1 = 88.6933426414
h2 = -44.3466713+270.9241255i
h3 = -44.3466713-270.9241255i

Calculated by our simple equation calculator.

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