To find the area of plastic wrap needed, we need to find the surface area of the triangular prism.
The triangular prism has 5 faces: 2 triangular ends and 3 rectangular sides.
First, let's find the area of the triangular ends. The dimensions given are 12 cm, 5 cm, 9 cm, and 13 cm. The triangular ends have dimensions of 5 cm and 12 cm (base and height). The area of a triangle is (1/2) * base * height.
Area of triangular ends = (1/2) * 5 cm * 12 cm = 30 cm². There are two of them, so the total area for both is 2 * 30 cm² = 60 cm².
Next, we need to find the area of the 3 rectangular sides. We know the longest dimension (13 cm) is the length of one side. For the other two sides, one has dimensions 12 cm and 13 cm, while the other has dimensions 5 cm and 13 cm.
Area of side 1: 12 cm * 13 cm = 156 cm²
Area of side 2: 5 cm * 13 cm = 65 cm²
There are two of side 1 and one of side 2, so the total for rectangular sides is 2 * 156 cm² + 65 cm² = 312 cm² + 65 cm² = 377 cm².
Finally, we add the areas of the triangular ends and rectangular sides to find the total surface area: 60 cm² + 377 cm² = 437 cm². Therefore, 437 cm² of plastic wrap is needed to cover the wedge of cheese.