Final answer:
To show that A and B have the same volume, we calculate the volume of both prisms. They do not have the same volume. To calculate the value of x for which D has the same volume as A, we equate their volumes and solve for x. The prism with the largest surface area is prism A.
Step-by-step explanation:
To show that A and B have the same volume, we need to calculate the volume of both prisms and compare them. The volume of a prism is given by the formula V = base area x height. For prism A, the base area is 20 cm x 20 cm = 400 cm² and the height is 25 cm. So, the volume of A is 400 cm² x 25 cm = 10,000 cm³. For prism B, the base area is 20 cm x 20 cm = 400 cm² and the height is 20 cm. So, the volume of B is 400 cm² x 20 cm = 8,000 cm³. Since the volume of A is greater than the volume of B, we can conclude that A and B do not have the same volume.
To calculate the value of x for which D has the same volume as A, we can equate their volumes and solve for x. The volume of A is 10,000 cm³ and the volume of D is (20 cm + 2x)³ cm³. Setting these equal, we have 10,000 = (20 + 2x)³. Simplifying and solving for x, we find x = 5 cm.
Since the radius of D is the same as the value calculated in (2), we know that the radius = 5 cm. To determine which prism will have the largest surface area, we need to find the surface area of each prism and compare them. The surface area of a prism is given by the formula SA = 2(base area) + (perimeter of base) x height. For prism A, the base area is 400 cm² and the perimeter of the base is 2(20 cm) + 2(20 cm) = 80 cm. The height is 25 cm. So, the surface area of A is 2(400 cm²) + (80 cm) x 25 cm = 9,600 cm². For prism B, the base area is 400 cm² and the perimeter of the base is 2(20 cm) + 2(20 cm) = 80 cm. The height is 20 cm. So, the surface area of B is 2(400 cm²) + (80 cm) x 20 cm = 8,800 cm². Since the surface area of A is greater than the surface area of B, prism A will have the largest surface area.
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