Question

aliyah is repainting a toy robot made out of two wooden cubes. she wrote the expression 6(15·15)+6(9·9)−2(9·9) for the surface area of the toy. in the expression aliyah wrote, what does the term 2(9·9) represent? the area of the overlap of the cubes why did aliyah subtract 2(9·9)? 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:

Sub - question 1: What does the term \(2(9\cdot9)\) represent?

The surface area of a cube is \(6\times(\text{side length}\times\text{side length})\). For the two cubes, when they are joined (overlapped), the overlapping area is covered on both cubes (one face from each cube). The area of one face of the smaller cube (with side length 9) is \(9\times9\), and since there are two faces (one from each cube) that are overlapped and not part of the total surface area of the toy, the term \(2(9\cdot9)\) represents the area of the overlap of the cubes (the area that is covered and should be subtracted from the sum of the surface areas of the two separate cubes).

When calculating the surface area of the toy (which is made by joining two cubes), if we just add the surface areas of the two individual cubes (\(6(15\cdot15)+6(9\cdot9)\)), we are including the area where the cubes overlap twice (once for each cube). But in reality, this overlapping area is not part of the surface area of the toy (because it is covered between the two cubes). So, we subtract \(2(9\cdot9)\) because the surface area of the toy should 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.

Answer:

The area of the overlap of the cubes

Sub - question 2: Why did Aliyah subtract \(2(9\cdot9)\)?