To find the surface area of a rectangular pyramid, you need to consider both the area of the rectangular base and the areas of the triangular faces.

Step 1: Area of the Base

The base is a rectangle with dimensions 4 ft by 4 ft. The area of the rectangle is:

$\text{Area of base} = \text{length} \times \text{width} = 4 \times 4 = 16 \, \text{square feet}$

Step 2: Area of the Triangular Faces

The pyramid has four triangular faces. Since the base is a square, two triangles will have a base of 4 ft and the other two will also have a base of 4 ft. The slant height of each triangle is 6 ft.

The area of one triangle is given by:

$\text{Area of one triangle} = \frac{1}{2} \times \text{base} \times \text{slant height}$

For each triangular face, we have:

$\text{Area of one triangle} = \frac{1}{2} \times 4 \times 6 = 12 \, \text{square feet}$

Since there are four triangles, the total area of the triangular faces is:

$\text{Total area of triangular faces} = 4 \times 12 = 48 \, \text{square feet}$

Step 3: Total Surface Area

Now, add the area of the base and the area of the four triangular faces:

$\text{Surface Area} = \text{Area of base} + \text{Total area of triangular faces}$ $\text{Surface Area} = 16 + 48 = 64 \, \text{square feet}$

Final Answer:

The surface area of the rectangular pyramid is 64 square feet.

Would you like a breakdown of any step or more details on any specific part of the problem?

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Here are some related questions:

1. How would the surface area change if the pyramid had a different slant height?
2. How can the volume of this pyramid be calculated?
3. What is the relationship between the slant height and the height of a pyramid?
4. How would you calculate the surface area of a non-rectangular pyramid (e.g., triangular or hexagonal)?
5. What are the differences between the surface area and volume formulas for pyramids?

Tip: To visualize the surface area, draw the pyramid and label all sides, including the base and the slant heights of the triangles.