Final answer:
To find out how many congruent cubes Richard can produce by cutting a rectangular prism, we need to find the dimensions of the cubes and divide the volume of the prism by the volume of each cube.
Step-by-step explanation:
To find out how many congruent cubes Richard can produce by cutting a rectangular prism, we need to find the dimensions of the cubes. The largest possible cubes that can be created would have sides that are equal to the greatest common divisor (GCD) of the lengths, widths, and heights of the prism.
To find the GCD of 4 1/2 ft, 7 1/2 ft, and 11 1/4 ft, we need to convert them to a common fractional form. The GCD of 4 1/2, 7 1/2, and 11 1/2 is 1/2 ft. Therefore, the side length of each cube will be 1/2 ft.
To find the number of cubes that can be produced, we need to divide the total volume of the rectangular prism by the volume of each cube. The cube's volume is found by multiplying the length of each side of the cube three times. The volume of the prism is given by its length times its width times its height. Therefore, the number of cubes that Richard can produce is (4 1/2 ft x 7 1/2 ft x 11 1/2 ft) / ((1/2 ft)^3) = 1836.