Final answer:
To fill a rectangular prism measuring 4 ½ x 6 ½ x 2 ½ inches with ½-inch cubes, calculate the volume of the prism and the cube. Then, divide the prism's volume by the cube's volume to find that 585 cubes are needed.
Step-by-step explanation:
To determine how many ½-inch cubes are needed to fill a rectangular prism with dimensions of 4 ½ inches long, 6 ½ inches wide, and 2 ½ inches high, we must calculate the volume of the prism and the volume of a single cube, and then find out how many cubes fit into the prism.
First, calculate the prism's volume:
Volume of prism = length × width × height
Volume of prism = 4.5 in. × 6.5 in. × 2.5 in. = 73.125 in.³
Next, calculate the volume of a cube:
Volume of cube = side × side × side
Volume of cube = 0.5 in. × 0.5 in. × 0.5 in. = 0.125 in.³
Finally, divide the prism's volume by the volume of a cube to find the number of cubes needed:
Number of cubes = Volume of prism / Volume of cube
Number of cubes = 73.125 in.³ / 0.125 in.³ = 585 cubes
Therefore, a total of 585 ½-inch cubes are required to fill the given rectangular prism.