Herga makes boxes for games that she has invented. she makesa box for one game. then she makes a box with dimensions that are 1/3 the dimensions of the first box. what is the relationship between the surface areas of the boxes ? explain.

Given that the first box has height h, width w, and length l, the surface area of a rectangular prism is A = 2wl + 2lh + 2hw. Since Herga makes another box with 1/3 the dimensions of the first, we multiply each variable by 1/3 to find the second surface area. So the surface area of the second box is A = 2(w/3)(l/3) + 2(l/3)(h/3) + 2(h/3)(w/3). This results in A = (1/9)(2wl + 2lh + 2hw). We can see that the surface area of the second box is 1/9 the surface area of the first box.

Call a, b and c the dimensions of the original box. Then the surface area is 2ab + 2bc + 2ac (because there are two sides of each type ab, bc, and ac.

The second box will have dimensions a/3, b/3 and c/3. So the surface area will be: 2(a/3)(b/3) + 2(b/3)(c/3) + 2(a/3)(c/3) = [1/9] {2ab + 2bc + 2ac}.

As you can see, the surface area of the first box is 9 times the surface area of the second box.

    area of the first box
= 9
area of the second box

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