Volume of the solid of revolution is 127.47 cubic units
What is Isosceles triangle?
An isosceles triangle is a triangle that has two sides of equal length.
What is Volume?
Volume is a scalar quantity expressing the amount of three-dimensional space enclosed by a closed surface.
What is Solid of revolution?
A solid of revolution is a solid figure obtained by rotating a plane curve around some straight line that lies on the same plane
Angles of the given isosceles triangle = 30-30-120 triangle
Length of the leg = 4 units
Length of the Third side c =[tex]\sqrt{a^{2}+b^{2}-2ab cos C }[/tex]
Where a and b are the length of the leg and C is the vertex angle
Then,
Length of the third side c= [tex]\sqrt{4^{2}+4^{2}-2(4)(4)cos120 } =4\sqrt{3}[/tex]
It is rotated around an axis that contains one leg
Draw the figure,
From the figure we can say that we have to subtract the volume of the big cone from the volume of the small cone to find the required volume of the cone
Volume of big cone = [tex]\frac{1}{3}\pi r^{2}\sqrt{l^{2}.r^{2} }[/tex]
where, r is the radius and l is the slant length
r = [tex]\frac{4\sqrt{3} }{2}=2\sqrt{3}[/tex] unit
l=[tex]4\sqrt{3}[/tex] unit
Volume of the big cone = [tex]\frac{1}{3}\pi (2\sqrt{3})^{2}\sqrt{(4\sqrt{3})^{2}. (2\sqrt{3})^{2} }[/tex]=301.59 cubic unit
Volume of small cone = [tex]\frac{1}{3}\pi r^{2}\sqrt{l^{2}.r^{2} }[/tex]
r = [tex]\frac{4\sqrt{3} }{2}=2\sqrt{3}[/tex] unit
l= 4 unit
Volume of the small cone = [tex]\frac{1}{3}\pi (2\sqrt{3})^{2}\sqrt{(4^{2} . (2\sqrt{3})^{2} }[/tex]=174.12 cubic unit
Volume of required region = Volume of big cone - Volume of small cone
=301.59 - 174.12
=127.47 cubic unit
Hence, volume of the solid of revolution is 127.47 cubic unit
Learn more about Isosceles triangle, Volume and Solid of revolution here
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