Answer to Question #291639 in Geometry for Marie Delos santos

Question #291639

The lateral surface area of a given right circular cylinder is twice its volume. If the height of the cylinder is twice its base diameter, determine the volume of the largest sphere you can put on the cylinder. Determine also the difference of the volume of the cylinder and sphere. 


1
Expert's answer
2022-01-31T13:31:58-0500

For a right circular cylinder of radius rr and height h,h, the lateral area is A=2πrhA=2\pi rh and the volume is V=πr2hV=\pi r^2 h

The lateral surface area of a given right circular cylinder is twice its volume


2πrh=2πr2h=>r=1unit2\pi rh=2\pi r^2h=>r=1 unit

The height of the cylinder is twice its base diameter


h=4r=4unitsh=4r=4 units

The volume of the largest sphere we can put on the cylinder is

Vsphere=43πr3=43π(1)3=4π3(units3)V_{sphere}=\dfrac{4}{3}\pi r^3 =\dfrac{4}{3}\pi (1)^3 =\dfrac{4\pi}{3}({units}^3)

The volume of the cylinder is


Vcyl=π(1)2(4)=4π(units3)V_{cyl}=\pi(1)^2(4)=4\pi ({units}^3)

The difference of the volume of the cylinder and sphere is


ΔV=VcylVsphere=4π4π3\Delta V=V_{cyl}-V_{sphere}=4\pi -\dfrac{4\pi}{3}

=8π3(units3)=\dfrac{8\pi}{3}({units}^3)


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