The total area of all the faces of a rectangular prism is 22, and the total length of all its edges is 24. Find the length of the internal diagonal of the prism.

Thank you!

I know I'm late, but...

I'm also getting √14.
~Dog_Lover

let the sides be a, b, and c

We know:
4a +4b + 4c = 24 ----> a+b+c = 6
and
4ab + 4ac + 4bc = 22
2ab + 2ac + 2bc = 11

The length of the diagonal is
√(a^2 + b^2 + c^2)

by algebra, we know
(a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc
subbing in
6^2 = a^2 + b^2 + c^2 +11
a^2 + b^2 + c^2 = 25
then ..

√(a^2 + b^2 + c^2) = √25 = 5

The diagonal is 5

(nice problem)

If the dimensions are x,y,z then we have

2xy+2xz+2yz = 22
4x+4y+4z = 24

xy+xz+yz = 11
x+y+z = 6

How about 1,2,3
diagonal is √14

Let the sides be a, b, and c. Then the total surface area is 2(ab+bc+ca)=22, and total length of all the edges is 4(a+b+c)=24. Hence, ab + ac + bc = 11 and a + b + c = 6.

We want to find the length of the internal diagonal, which is √a^2 + b^2 + c^2. First, we square the equation a + b + c = 6, to get
a^2 + b^2 + c^2 + 2ab + 2ac + 2bc = 36.

Therefore,
a^2 + b^2 + c^2 = 36 - 2(ab + ac + bc) = 36 - 2 * 11 = 14,
which means that the length of the internal diagonal is √14.

Sorry, tried 5 but it didn't work. The hint said that it was a simplified square root?

How did I possible see 4 of equal sides ?

but... my algebra worked out sooo nice!

Reminds me of the classic StarTrek episode where Cpt Picard is tortured by the Kardashians and sees 4 lights.
https://www.youtube.com/watch?v=moX3z2RJAV8

To find the length of the internal diagonal of a rectangular prism, we can use the given facts about its surface area and total edge length.

Let's start by understanding the properties of a rectangular prism. A rectangular prism has six faces, with each pair of opposite faces being congruent. The opposite faces are also parallel.

To find the total area of all the faces, we sum up the area of each face. Since a rectangular prism has six faces, we can represent the area of all the faces as:

Total Area = 2(ab + ac + bc)

where a, b, and c represent the length, width, and height of the prism, respectively.

In this problem, the total area of all the faces is given as 22. So we have the equation:

22 = 2(ab + ac + bc) ----(1)

Next, let's consider the total length of all the edges. In a rectangular prism, each edge is shared by two faces. Therefore, the total length of all the edges is equal to the sum of the perimeters of all the faces.

Since a rectangular prism has 12 edges, we can write the equation for the total length of all the edges as:

Total Edge Length = 4a + 4b + 4c

In this problem, the total length of all the edges is given as 24. So we have the equation:

24 = 4a + 4b + 4c ----(2)

To find the length of the internal diagonal, we can use the Pythagorean theorem. The internal diagonal is the hypotenuse of a right-angled triangle formed by the three dimensions of the prism.

The Pythagorean theorem states that the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, the internal diagonal is the hypotenuse, and the sides of the triangle are the length (a), width (b), and height (c) of the prism.

So the equation can be written as:

Diagonal^2 = a^2 + b^2 + c^2

To solve this problem, we need to find the values of a, b, c, and the internal diagonal (Diagonal) that satisfy equations (1), (2), and the Pythagorean theorem equation.

Now we can solve the problem by finding the values of a, b, c, and the internal diagonal using simultaneous equations.