In this problem, Quincy folds a piece of cardboard to create three sides of a triangular prism. The prism is open at both the top and bottom. We need to calculate the area of the cardboard used.

Information provided:

• The triangular base has side lengths of 9 cm, 9 cm, and 7 cm.
• The height of the prism is 20 cm.

Steps:

1. Unfold the triangular prism: The unfolded cardboard forms a rectangle consisting of three rectangular faces, corresponding to the three sides of the triangular base.

2. Dimensions of the rectangular faces:

   • The length of each rectangle is the height of the prism (20 cm).
   • The width of each rectangle is the side length of the triangular base.
     • One rectangle has a width of 9 cm.
     • The second rectangle has a width of 9 cm.
     • The third rectangle has a width of 7 cm.

3. Calculate the area of each rectangle:

   • Area of the first rectangle: $9 \, \text{cm} \times 20 \, \text{cm} = 180 \, \text{cm}^2$
   • Area of the second rectangle: $9 \, \text{cm} \times 20 \, \text{cm} = 180 \, \text{cm}^2$
   • Area of the third rectangle: $7 \, \text{cm} \times 20 \, \text{cm} = 140 \, \text{cm}^2$

4. Total area of the cardboard: Add the areas of all three rectangles: $180 \, \text{cm}^2 + 180 \, \text{cm}^2 + 140 \, \text{cm}^2 = 500 \, \text{cm}^2$

Thus, the area of the piece of cardboard is 500 cm².

Would you like further clarification or details? Here are some related questions:

1. How do you calculate the surface area of a complete triangular prism (including top and bottom faces)?
2. What is the formula for the area of a triangle, and how would it apply if the prism had top and bottom covers?
3. How does the shape of the base affect the surface area of a prism?
4. What are other common shapes for prisms, and how do their surface areas compare to triangular prisms?
5. How would the area change if the height of the prism were doubled?

Tip: Understanding how to decompose 3D shapes into simpler 2D parts can make surface area calculations much easier!