A rectangular box $P$ is inscribed in a sphere of radius $r$. The surface area of $P$ is 384, and the sum of the lengths of its 12 edges is 112. What is $r$? $\mathrm{(A) } 8 \qquad$ $ \mathrm{(B) } 10 \qquad$ $ \mathrm{(C) } 12 \qquad$ $ \mathrm{(D) } 14 \qquad$ $ \mathrm{(E) } 16$
Solution
Let the edges be $a,b,c$. Given $4(a+b+c)=112\Rightarrow a+b+c=28$, and $2(ab+bc+ca)=384$.
The box's space diagonal (the sphere's diameter) satisfies $(2r)^2=a^2+b^2+c^2=(a+b+c)^2-2(ab+bc+ca)=784-384=400$.
A rectangular box $P$ is inscribed in a sphere of radius $r$. The surface area of $P$ is $320$, and the sum of the lengths of its $12$ edges is $96$. What is $r$?
$\mathrm{(A) } 8 \qquad$ $ \mathrm{(B) } 10 \qquad$ $ \mathrm{(C) } 12 \qquad$ $ \mathrm{(D) } 14 \qquad$ $ \mathrm{(E) } 16$ 