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# Prism

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A rectangular prism has a total surface area of 56. Also, the sum of all the edges of the prism is $$56$$. Find the length of the diagonal joining one corner of the prism to the opposite corner.

May 13, 2022

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So, let $$h \times w \times l$$ be the dimensions of that prism.

We have $$\begin{cases}2(hw + wl + hl) = 56\\h + w + l = 56\end{cases}$$. We are to find $$\sqrt{h^2 + w^2 + l^2}$$.

We can use the identity $$(h + w + l)^2 = (h^2 + w^2 + l^2) + 2(hw + wl + hl)$$ to calculate $$h^2 + w^2 + l^2$$, and then take square root. I will leave that as an exercise for you.

If you don't believe the identity is true, try to expand the left hand side and the right hand side. $$(h + w + l)^2 = ((h + w) + l)^2 = (h + w)^2 + 2l(h + w) + l^2$$ and etc.

May 14, 2022