Question

aliyah is repainting a toy robot made out of two wooden cubes. she wrote the expression $6(15 cdot 15)+6(9 cdot 9)-2(9 cdot 9)$ for the surface area of the toy. in the expression aliyah wrote, what does the term $2(9 cdot 9)$ represent? the area of the overlap of the cubes why did aliyah subtract $2(9 cdot 9)$? the surface area of the toy? include the area where the cubes overlap. as a result, the surface area of the toy is? the sum of the surface areas of both cubes.

Explanation:

Step1: Analyze the surface - area formula for combined solids

When two cubes are joined, the overlapping parts are not part of the surface area. Each cube has \(6\) faces. The formula for the surface area of a single cube is \(6s^{2}\), where \(s\) is the side - length of the cube. For two separate cubes, the total surface area would be \(6s_1^{2}+6s_2^{2}\) (where \(s_1 = 15\) and \(s_2=9\)). But when the two cubes are joined, two faces (one from each cube) are overlapping. The area of each of these overlapping faces is \(9\times9\) (since the side - length of the smaller cube is \(9\)).

Step2: Relate the formula to the problem

The term \(2(9\times9)\) represents the area of the two overlapping faces. When we calculate the surface area of the combined solid (the toy robot), we need to subtract the area of the overlapping parts from the sum of the surface areas of the two individual cubes.

Answer:

The surface area of the toy does not include the area where the cubes overlap. As a result, the surface area of the toy is less than the sum of the surface areas of both cubes.