A rectangular prism has a total surface area of Also, the sum of all the edges of the prism is Find the length of the diagonal joining one corner of the prism to the opposite corner.
Sorry I don't know if it's just me, but please check your formatting when you post a question. I can't see any values or numbers that are supposed to be in the problem.
A rectangular prism has a total surface area of 48. Also, the sum of all the edges of the prism is 40. Find the length of the diagonal joining one corner of the prism to the opposite corner.
Let's call the lengths of the rectangular prism x, y, and z. The problem tells us that the sum of the edges is 40, so we can write:
x + y + z = 40
The problem also says that the surface area is 48. That means:
2(xy +yz + xz) = 48
Realize that if we square (x + y + z), that is equal to :
(x^2 + y^2 + z^2 + 2xy + 2xz + 2yz), or :
(x^2 + y^2 + z^2) + 2(xy + xz + yz)
(try expanding it out yourself if you want!)
(x^2 + y^2 + z^2 ) + 48 (since the surface area is 48. See how the substitution works?).
This is then equal to 1600(40 ^2).
Subtracting 48 on both sides, we get:
(x^2 + y^2 + z^2) = 1552
What's important to realize here is that the question asks us for the space diagonal(the diagonal connecting opposite corners of the prism). The general formula for the length of a space diagonal in a rectangular prism is: sqrt(x^2 + y^2 + z^2). This can be derived from using pythagorean theorem if you really want to see for yourself.
That means our desired diagonal length is sqrt(1552), which is simplifies to : 4sqrt(97)