To decompose the box net into simple polygons and determine the area of cardboard needed to make the cube, we start by visualizing the box (or cube) based on the dimensions provided.
From the dimensions you’ve given (4 in, 8 in, 4 in, 4 in, 8 in), it appears that the box is a rectangular prism (not a cube, since it has different lengths). Let's clarify the dimensions first:
- Two dimensions are given as 8 inches (possibly the length and width).
- Two dimensions are given as 4 inches (possibly the height of the box).
Thus, we have a rectangular prism (box) with:
- Length (L) = 8 inches
- Width (W) = 4 inches
- Height (H) = 4 inches
Next, the surface area of a rectangular prism can be calculated using the formula:
\[ \text{Surface Area} = 2(LW + LH + WH) \]
Now, plug in the values:
- \(L = 8\)
- \(W = 4\)
- \(H = 4\)
Let's compute each term:
- \(LW = 8 \times 4 = 32\) square inches
- \(LH = 8 \times 4 = 32\) square inches
- \(WH = 4 \times 4 = 16\) square inches
Now, sum these areas: \[ LW + LH + WH = 32 + 32 + 16 = 80 \text{ square inches} \]
Now, multiply this sum by 2 to find the total surface area: \[ \text{Surface Area} = 2 \times 80 = 160 \text{ square inches} \]
Thus, the area of cardboard needed to create the rectangular prism (box) is 160 square inches.