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# Geometry

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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 $$62$$. Find the length of the diagonal joining one corner of the prism to the opposite corner.

Apr 20, 2022

#1
+1370
+2

We have: $$4x+4y+4z=62$$ and $$2xy+2yz+2xz=56$$

Simplifying the first equation, we get: $$x+y+z=15.5$$

Recall that $$(x+y+z)^2=x^2+y^2+z^2+2xy+2yz+2xz$$

Substituting what we know, we have: $$x^2+y^2+z^2+56=240.25$$

We can transform this equation to: $$x^2+y^2+z^2=184.25$$

The length of the diagonal is: $$\sqrt{x^2+y^2+z^2}$$

Can you find the answer now?

Apr 20, 2022

#1
+1370
+2

We have: $$4x+4y+4z=62$$ and $$2xy+2yz+2xz=56$$

Simplifying the first equation, we get: $$x+y+z=15.5$$

Recall that $$(x+y+z)^2=x^2+y^2+z^2+2xy+2yz+2xz$$

Substituting what we know, we have: $$x^2+y^2+z^2+56=240.25$$

We can transform this equation to: $$x^2+y^2+z^2=184.25$$

The length of the diagonal is: $$\sqrt{x^2+y^2+z^2}$$

Can you find the answer now?

BuilderBoi Apr 20, 2022