Bunuel wrote:
What is the volume of a given cube?
rubb1ees wrote:
Can someone please help explain the answer
How can do you find the sides from the diagonal
(1) The ratio of an edge of the cube and the greatest distance between two points on the cube is \(1: \sqrt{3}\).
(2) The length of the diagonal across a face of the cube is 2.
IF YOU FIND MY SOLUTION HELPFUL, PLEASE GIVE ME KUDOS
(1). The ratio of an edge of a cube and the greatest distance between two points on the cube is 1:root3. This actually provides no information. When we have a cube, diagonal of a face is root2*side length, and the greatest distance between two points = sqrt (side length^2 +(side length *root2)^2) = sqrt( side length^2 + 2 side length squared)= sqrt(3*side length squared) = side length * root 3 per the pythagorean theorem. picking values for the side length will give us corresponding values for the diagonal of the cube. For example if side length = 1, we have a voume of 1^3 = 1 with face diagonal = 1*root2 and cube diagonal 1*root 3, but if our side length = 2 we have volume 2^3 = 8, with face diagonal 2*root 2 and cube diagonal 2*root3 NS
(2) The length of the diagonal across a face is 2. Per the pythagorean theorem, any square face with diagonal 2 will be found by solving 2 = 2x^2 --> x =1 (since the diagonal and 2 of the 4 sides make a right triangle. Since all sides of a cube are equal, both sides can be represented with x, thus x^2+x^2 =2^2 per Pythagorean theorem)
Thus, our volume = 1^3 = 1. Sufficient
The answer is B