Question

Express the edge length of a cube as a function of the cube's diagonal length d. Then express the surface area & volume of the cube as a function of the diagonal length.

Answer 1

To express the edge length of a cube as a function of the cube's diagonal length, use the Pythagorean theorem. The surface area of the cube is 6 times the edge length squared, and the volume is the edge length cubed.

Step-by-step explanation:

To express the edge length of a cube as a function of the cube's diagonal length, we can use the Pythagorean theorem. Let's assume the edge length is 'a' and the diagonal length is 'd'.

sing the Pythagorean theorem in a right triangle formed by the edge length, diagonal length, and height of the cube, we have:

a² + a² + a² = d²

3a² = d²

a = √(d²/3)

To express the surface area of the cube as a function of the diagonal length, we can use the formula for the surface area of a cube, which is 6 times the edge length squared:

Surface Area = 6a²

To express the volume of the cube as a function of the diagonal length, we can use the formula for the volume of a cube, which is the edge length cubed

Volume = a³

Answer 2

If we consider "a" as the edge length, and "D" the cube's diagonal, we have that the square cube's diagonal is equal to the edge length's square plus the side diagonal (d) square (Pythagoras theorem)

a² + d² = D²

And since:
d² = a² + a²

Clearing a, we have:
a² = D²-d²
a² = D²-2a²
3a² = D²
a = √(D²/3)

Surface area is equal to 6·a², so the surface area will be 6·(D²/3) = 2D²

The volume is a³, so the volume will be √(D²/3)³ = √(
`D^(6)`/3³) = D³/√27