Question

the edge length, in millimeters (mm), of cube y is 189 times the edge length, in mm, of cube x. the surface area, in mm², of cube y is n times the surface area, in mm², of cube x. what is the value of n? a 189 b 1,134 c 35,721 d 214,326

Explanation:

Step1: Recall the surface - area formula for a cube

The surface area formula of a cube is \(S = 6s^{2}\), where \(s\) is the edge - length of the cube. Let the edge - length of cube \(X\) be \(s_{X}\), then the surface area of cube \(X\) is \(S_{X}=6s_{X}^{2}\). The edge - length of cube \(Y\) is \(s_{Y}=189s_{X}\), and the surface area of cube \(Y\) is \(S_{Y}=6s_{Y}^{2}\).

Step2: Substitute \(s_{Y}=189s_{X}\) into the surface - area formula for cube \(Y\)

Substitute \(s_{Y}\) into \(S_{Y}\): \(S_{Y}=6(189s_{X})^{2}\). Using the formula \((ab)^{2}=a^{2}b^{2}\), we get \(S_{Y}=6\times189^{2}s_{X}^{2}\). Since \(S_{X}=6s_{X}^{2}\), then \(\frac{S_{Y}}{S_{X}}=\frac{6\times189^{2}s_{X}^{2}}{6s_{X}^{2}}\).

Step3: Simplify the ratio \(\frac{S_{Y}}{S_{X}}\)

Cancel out the common factors \(6\) and \(s_{X}^{2}\) in the fraction \(\frac{6\times189^{2}s_{X}^{2}}{6s_{X}^{2}}\). We are left with \(n = 189^{2}\). Calculate \(189^{2}=(190 - 1)^{2}=190^{2}-2\times190\times1 + 1^{2}=36100-380 + 1=35721\) (or \(189\times189=(200 - 11)\times(200 - 11)=40000-4400 + 121=35721\)).

Answer:

C. \(35,721\)