Question

Anya is wrapping gift boxes in paper. Each gift box is a rectangular prism. The larger box has a length, width, and height twice as large as the smaller box. Which statement shows the relationship between the surface areas of the gift boxes?A. The larger gift box has a surface area 2 times as large as the smaller gift box.B. The larger gift box has a surface area 4 times as large as the smaller gift box.C. The larger gift box has a surface area 6 times as large as the smaller gift box.D. The larger gift box has a surface area 8 times as large as the smaller gift box.thank you ! :)

Answer 1

The surface area of a rectangular prism is calculated using the formula:

`\begin{gathered} SA=2LW+2LH+2WH \\ or \\ SA=2(LW+LH+WH) \end{gathered}`

Let us assume that this is the surface area of the smaller box.

Now, let's represent the surface area of the bigger box by doubling the dimensions of the smaller box.

`SA_(big)=2(2L\cdot2W+2L\cdot2H+2W\cdot2H)`

Let's simplify this by factoring out the common factors.

`\begin{gathered} SA_(b\imaginaryI g)=2(2L\cdot2W+2L\cdot2H+2W\cdot2H) \\ \\ SA_(b\imaginaryI g)=2\cdot2\cdot2(L\cdot W+L\cdot H+W\cdot H) \\ \\ SA_(big)=4\cdot2(LW+LH+WH) \\ \\ SA_(big)=4SA \end{gathered}`

So the answer is option B (The larger gift box has a surface area 4 times as large as the smaller gift box).