To calculate the amount of wrapping paper used, we need to find the surface area of the net of the triangular prism. The net consists of:

1. Two triangular bases.
2. Three rectangular faces.

Step-by-Step Process:

1. Triangular Bases: The base of each triangle has dimensions:

   • Base = 6 ft
   • Height = 5 ft (as per the dotted line)

   The area of one triangle is given by the formula: $A_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6 \times 5 = 15 \text{ square feet}$ Since there are two triangular bases: $\text{Total area of triangles} = 2 \times 15 = 30 \text{ square feet}$

2. Rectangular Faces: The three rectangles have different dimensions:

   • The large rectangle has dimensions: $7 \text{ ft (length)} \times 6 \text{ ft (width)} = 42 \text{ square feet}$

   • The two smaller rectangles attached to the triangular faces each have dimensions: $7.81 \text{ ft (length)} \times 6 \text{ ft (width)} = 46.86 \text{ square feet}$ Since there are two such rectangles: $\text{Area of two rectangles} = 2 \times 46.86 = 93.72 \text{ square feet}$

3. Total Surface Area: Adding the areas of the triangles and rectangles: $\text{Total area} = 30 + 42 + 93.72 = 165.72 \text{ square feet}$

Thus, Josiah used approximately 165.72 square feet of wrapping paper.

Do you have any questions or would you like further details?

Related Questions:

1. How do you calculate the surface area of a cylinder-shaped object?
2. What is the formula for the volume of a triangular prism?
3. How does the surface area of a prism change with different dimensions?
4. How would the wrapping paper amount change if the prism had different angles in the triangular bases?
5. What other shapes can be used to calculate surface area for packaging purposes?

Tip:

When calculating surface areas, always break the shape into simpler parts (like rectangles and triangles), solve individually, and then sum them up!