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Maximum volume of a rectangular cuboid inscribed in a sphere

ÀÛ¼ºÀÚ Uploader : pleiades ÀÛ¼ºÀÏ Upload Date: 2019-11-22º¯°æÀÏ Update Date: 2022-06-18Á¶È¸¼ö View : 85

Find the maximum volume of a rectangular cuboid inscribed in a sphere of radius r.

If the width, length, and height of the rectangular cuboid are a, b, and c, respectively, the volume V of the rectangular cuboid is:

V = a*b*c

Since the diagonal length of the rectangular cuboid is the diameter of the sphere,

2r = (a^2 + b^2 + c^2)^(1/2)

4r^2 = a^2 + b^2 + c^2

c^2 = 4r^2 - a^2 - b^2

Thus,

V^2 = (a*b*c)^2

    = a^2*b^2*(4r^2 - a^2 - b^2)

    = a^2*b^2*(4r^2 - (a^2 + b^2))

Using the relationship between the arithmetic mean and the geometric mean, (a^2+b^2)/2 ¡Ã a*b,

When a=b, a^2*b^2 is the maximum and a^2+b^2 is the minimum, so V becomes the maximum.

V^2 = a^4*(4r^2-2a^2) = 4r^2a^4 - 2a^6

Differentiating with respect to a,

4r^2*4a^3 - 6*2a^5 = 0

3a^2 = 4r^2

a = (2/3^(1/2))*r

Therefore,

Vmax = (8/(3*3^(1/2)))r^3

*** Âü°í¹®Çå[References] ***

Vmax = (8/(3*3^(1/2)))r^3
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