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clavoie
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.
Each side of the cube has length x On one face of the cube, the diagonal is found using Pythagorean's Theorem a^2 + b^2 = c^2 Since this is a cube, a= b = x. And c^2 = sqrt(x^2 + x^2) , so c= sqrt(2x^2) Now d, the diagonal of the cube, can be found side it is formed by a triangle with a side of length x and a side of length sqrt(2x^2) So d^2=( sqrt(2x^2))^2 + x^2 d = sqrt(3x^2) Solving for x gives x=sqrt((d^2)/3) Each face has an area of x^2 = (d^2)/3 Surface area = 6 x ((d^2)/3). ( there are 6 faces on a cube) And volume is x^3 = (sqrt((d^2)/3))^3 That was not easy to type up on an iPad!