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# Geometry

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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.

Jul 21, 2024

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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 $$\frac{7}{4}$$ :)

Jul 21, 2024