The ratio of side lengths for two similar cubes is 2/5. Determine the ratios for each of the following. Show your work or explain how you got your answers.
a. the perimeter of one face of the smaller cube compared to the perimeter of one face of the larger cube
b. the total surface area of the smaller cube compared to the total surface area of the larger cube
c. the volume of the smaller cube compared to the volume of the larger cube
Let x be the length of a side of the smaller cube, and let y be the length of a side of the larger cube.
a. The perimeter of one face of a cube is simply 4 times the length of a side. So the ratio of the perimeters of the faces of the smaller cube to the larger cube is 4x/4y = x/y.
b. The surface area of a cube is given by 6 times the square of the length of a side. So the ratio of the total surface areas of the smaller cube to the larger cube is 6x^2 / 6y^2 = x^2 / y^2.
c. The volume of a cube is given by the cube of the length of a side. So the ratio of the volumes of the smaller cube to the larger cube is x^3 / y^3.
Therefore, the ratios are:
a. x/y
b. x^2/y^2
c. x^3/y^3