Let's calculate the surface area and volume for both the pyramid and the prism and compare the results.
a. Surface Area:
The surface area of a regular pyramid is given by the formula:
SA_pyr = Base Area + (1/2) * Perimeter of Base * Slant Height
The base area of the pyramid is the area of a square, which is equal to one side length squared:
Base Area_pyr = (10 inches)² = 100 square inches
The perimeter of the base is the sum of all four sides:
Perimeter of Base_pyr = 4 * 10 inches = 40 inches
Plugging the values into the formula, we have:
SA_pyr = 100 square inches + (1/2) * 40 inches * 13 inches
SA_pyr = 100 square inches + 20 inches * 13 inches
SA_pyr = 100 square inches + 260 square inches
SA_pyr = 360 square inches
Now, let's calculate the surface area of the prism. The surface area of a prism with a square base is given by the formula:
SA_prism = 2 * Base Area + Perimeter of Base * Height
The base area of the prism is the area of a square, which is equal to one side length squared:
Base Area_prism = (10 inches)² = 100 square inches
Plugging the values into the formula, we have:
SA_prism = 2 * 100 square inches + 40 inches * 12 inches
SA_prism = 200 square inches + 480 square inches
SA_prism = 680 square inches
Therefore, the pyramid has a surface area of 360 square inches, and the prism has a surface area of 680 square inches.
The prism has a larger surface area by:
680 square inches - 360 square inches = 320 square inches.
b. Volume:
The volume of a regular pyramid is given by the formula:
V_pyr = (1/3) * Base Area * Height
Plugging the values into the formula, we have:
V_pyr = (1/3) * 100 square inches * 12 inches
V_pyr = 400 cubic inches
The volume of a prism with a square base is given by the formula:
V_prism = Base Area * Height
Plugging the values into the formula, we have:
V_prism = 100 square inches * 12 inches
V_prism = 1200 cubic inches
Therefore, the pyramid has a volume of 400 cubic inches, and the prism has a volume of 1200 cubic inches.
The prism has a larger volume by:
1200 cubic inches - 400 cubic inches = 800 cubic inches.
To summarize:
a. The prism has a larger surface area by 320 square inches.
b. The prism has a larger volume by 800 cubic inches.