Question

1. finally, calculate the surface area of the entire prism. show all of your steps and explain your reasoning for how you determined which faces to include. extension (dok 4): if the prism is painted on the outside and then cut into 1 cm³ cubes, how many of the small cubes will have paint on at least one face? explain your reasoning.

Explanation:

Step1: Analyze cube - prism relationship

The small cubes on the surface of the prism will have paint on at least one face. Cubes on the interior of the prism will have no paint.

Step2: Consider prism dimensions

Let the dimensions of the prism be \(l\times w\times h\) (length, width, height). The number of small cubes with at - least one face painted can be found by considering the non - interior cubes. The number of interior cubes forms a smaller prism with dimensions \((l - 2)\times(w - 2)\times(h - 2)\) (assuming \(l\geq2\), \(w\geq2\), \(h\geq2\)). The total number of \(1\mathrm{cm}^3\) cubes in the prism is \(V=l\times w\times h\). The number of cubes with at least one face painted \(N = lwh-(l - 2)(w - 2)(h - 2)\).

Step3: Calculate surface area of prism

The surface area formula of a rectangular prism is \(SA = 2(lw+lh + wh)\).

However, since the dimensions of the prism are not given in the problem, we can only provide the general methods. If we assume the prism has length \(l\), width \(w\), and height \(h\):

Answer:

The number of small cubes with paint on at least one face is \(lwh-(l - 2)(w - 2)(h - 2)\) (assuming \(l\geq2\), \(w\geq2\), \(h\geq2\)). The surface area of the prism is \(2(lw + lh+wh)\)