+1694 Let $ABCDEFGH$ be right rectangular prism. The total surface area of the prism $15.$ Also, the sum of all the edges of the prism is $17.$ Find the length of the diagonal joining one corner of the prism to the opposite corner.
+16 Let's assign the values of the dimensions of the prims as \(p, q, r.\)
For the surface area: \(2(pq + pr + qr) = 15\)
For the sides: \(p + q + r = \frac{17}{4}\)
To find the diagonal we have to know: \sqrt{p^2 + q^2 + r^2}
With some basic algebraic manipulation, we know that \((p+q+r)^2 = p^2 + q^2 + r^2 + 2(pq + pr + qr)\), therefore:
\(p^2 + q^2 + r^2 = (p+q+r)^2 - 2(pq + pr + qr)\). We can substitute the values we know and write,
\(p^2 + q^2 + r^2 = (\frac{17}{4})^2 - (15), p^2 + q^2 + r^2 = \frac{289}{16} - \frac{240}{16}, p^2 + q^2 + r^2 = \frac{49}{16}, \)
therefore \(\sqrt{p^2 + q^2 + r^2} = \sqrt{\frac{49}{16}} = \frac{7}{4}\).
Now we found out that the diagonal of the prism is \(\frac{7}{4}\) :)
