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# Sphere in cube

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A spherical holiday ornament with a radius r is to be packacked in a cubical box so that all sides touch the sphere. What is the ratio of the volume of the sphere to the volume of the cube? If the same ornament in the problem above is to be packed in a cylinder so that all surfaces touch the sphere, what is the ratio of the volume of the sphere to the volume of the cylinder?

Mellie  Apr 21, 2017
#1
+91160
+4

Good to see you, Mellie....!!!

V cube  =   s^3        where s is the side length of the cube   (1)

V sphere in a cube   =  (4/3)pi (1/2s)^3    (2)

Ratio  of  (2)  to (1)   =

(4/3)pi (1/2s)^3 /  s^3   =

(4/3)pi *(1/8)s^3 /s^3

(4/3)pi (1/8) / 1  =

pi / 6

V cylinder  =   pi * r^2 * h  →  pi*r^2 *(2 r )  → 2* pi*r^3  (1)

V sphere  =   (4/3) * pi * r^3    (2)

Ratio  of  (2)  to (1)  =

[(4/3) pi * r^3 ] / [ 2 *pi * r^3 ]  =

(4/3) /  2  =

4/6  =

2/3

CPhill  Apr 21, 2017
#2
+1794
+1

HI!!! I haven't been here in a while. It's good to see you too! Thank you so much for all your help!

Mellie  Apr 21, 2017