<p>Let <img alt="$\mathcal{B}$" height="13" src="https://latex.artofproblemsolving.com/3/8/2/382cea990fadd43376cf42e8e0597a5eb13a67c0.png" width="12" /> be the set of rectangular boxes with surface area <img alt="$54$" height="13" src="https://latex.artofproblemsolving.com/b/1/c/b1cb5bdce2d06d2419066a2b55ab01d824217bd2.png" width="18" /> and volume <img alt="$23$" height="12" src="https://latex.artofproblemsolving.com/9/a/1/9a1b2497e7b125cd721bab4c6e2385f954fe35cb.png" width="17" />. Let <img alt="$r$" height="8" src="https://latex.artofproblemsolving.com/b/5/5/b55ca7a0aa88ab7d58f4fc035317fdac39b17861.png" width="8" /> be the radius of the smallest sphere that can contain each of the rectangular boxes that are elements of <img alt="$\mathcal{B}$" height="13" src="https://latex.artofproblemsolving.com/3/8/2/382cea990fadd43376cf42e8e0597a5eb13a67c0.png" width="12" />. The value of <img alt="$r^2$" height="15" src="https://latex.artofproblemsolving.com/e/2/2/e2221c84f16a3f75ae07e5023384cc40d9c05357.png" width="14" /> can be written as <img alt="$\frac{p}{q}$" height="36" src="https://latex.artofproblemsolving.com/7/3/4/7347a1d5586e82c832f61ea799d89945ed2837fc.png" width="11" />, where <img alt="$p$" height="11" src="https://latex.artofproblemsolving.com/3/6/f/36f73fc1312ee0349b3f3a0f3bd9eb5504339011.png" width="10" /> and <img alt="$q$" height="11" src="https://latex.artofproblemsolving.com/0/6/1/0615acc3725de21025457e7d6f7694dab8e2f758.png" width="8" /> are relatively prime positive integers. Find <img alt="$p+q$" height="14" src="https://latex.artofproblemsolving.com/1/7/0/170200756543596efeb46fca1cf29a617e8c9192.png" width="41" />.</p>

Question No: 414

Let $\mathcal{B}$ be the set of rectangular boxes with surface area $54$ and volume $23$ . Let $r$ be the radius of the smallest sphere that can contain each of the rectangular boxes that are elements of $\mathcal{B}$ . The value of $r^2$ can be written as $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p+q$ .

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Tags: Geometry

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