Answer:
Step-by-step explanation:
Inicially we have to separate the sphere into 3 parts:
The donut + 1 ball cap (or a dome) + the cylinder hole = the Hemisphere
vide picture
As it shows in the picture we have to find the donut.
The donut = the Hemisphere - (
1 dome + the cylinder hole)
- The hemisphere = [tex]\frac{4*pi*r^{3} }{6}[/tex] (from the hemisphere formula)
- 1 dome* = [tex]\frac{1}{6}*pi*(r-\sqrt{(r^{2} -a^{2}})*(3a^{2}+((r-\sqrt{(r^{2} -a^{2}})^{2} )[/tex]
- cylinder hole = [tex]pi*a^{2}*h = pi*a^{2} *(2r - 2*(r-\sqrt{(r^{2} -a^{2}}))[/tex] (h:
cylinder height)
[tex]D = \frac{4*pi*r^{3} }{6} - (\frac{1}{6}*pi*(r-\sqrt{(r^{2} -a^{2}})*(3a^{2}+((r-\sqrt{(r^{2} -a^{2}})^{2} )+ pi*a^{2}*(2r - 2*(r-\sqrt{(r^{2} -a^{2}})) }[/tex]
[tex]D = \frac{4*pi*2^{3} }{6} - (\frac{1}{6}*pi*(2-\sqrt{(2^{2} -1^{2}})*(3*1^{2}+((2-\sqrt{(2^{2} -1^{2}})^{2} )+ pi*1^{2}*(2*2 - 2*(2-\sqrt{(2^{2} -1^{2}})) }[/tex]
By basic algebra we have
D = 5.304
*The dome in the 1st picture is represented by the color orange.Looking into the second picture, the dome formula derivatives from the volume of a dome of a hemisphere. By using the following linear system:
- triangle_side2 + a2 = r2
- h + triangle_side = r
