Question #148943
__17. A rectangular solid whose dimensions are 4cm, 8cm, and 12cm is inscribed in a sphere. What is the diagonal of the rectangular solid?

_18. In question 17, what is the surface area of the sphere in ¹?
1
Expert's answer
2020-12-16T08:11:30-0500

17. Let a=a= the length of a rectangular solid, b=b= its width, a c=c= its height.

The the diagonal dd of a rectangular solid will be


d=a2+b2+c2d=\sqrt{a^2 +b^2+c^2}d=42+82+122=414(cm)14.97(cm)d=\sqrt{4^2+8^2+12^2}=4\sqrt{14}(cm)\approx14.97(cm)



18. If the rectangular solid is inscribed in a sphere, the center of the sphere is the point of intersection of the diagonals of the rectangular solid. Then


Rsphere=12d=12(414cm)=214 cmR_{sphere}=\dfrac{1}{2}d=\dfrac{1}{2}(4\sqrt{14}cm)=2\sqrt{14}\ cm

The surface area of the sphere is


A=4πRsphere2=4π(214cm)2A=4\pi R_{sphere}^2=4\pi (2\sqrt{14}cm)^2=224π cm2703.72 cm2=224\pi\ cm^2\approx703.72\ cm^2




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