Answer:
(a) 1/2 inch
(b) 7/2 cubic inches
(c) 28
Step-by-step explanation:
You want the size and number of the largest unit cubes that will fill a rectangular prism with dimensions of 1/2, 2, and 7/2 inches. You also want the volume of the prism.
(a) Unit cube
We need to choose a unit cube whose edge length fits evenly into the prism edge lengths.
We notice that 2 and 7/2 are whole number multiples of 1/2. Hence 1/2 inch is the edge dimension of the largest unit cube that can be used to pack the prism.
(b) Volume
The volume of a rectangular prism is the product of its dimensions:
V = LWH
V = (1/2 in)(2 in)(7/2 in) = 7/2 in³
The volume of the prism is 7/2 = 3.5 cubic inches.
(c) Cubes
The total number of unit cubes required to fill the prism is the product of the number of them in each direction.
The 1/2-inch dimension accommodates 1 cube of length 1/2 inch.
The 2-inch dimension accommodates 4 cubes of length 1/2 inch.
The 7/2-inch dimension accommodates 7 cubes of length 1/2 inch.
The total number of cubes required to fill the prism is ...
1 × 4 × 7 = 28
28 cubes will fit into the prism.
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Additional comment
The size of the unit cube is the greatest common factor of 1/2, 2, and 7/2. There are several ways you can find this value. Using your knowledge of fractions and whole numbers, as above, is the easiest.
More formally, if you divide both of the larger dimensions by the smaller, and the remainders are 0, then the smaller dimension is the greatest common factor. 2 ÷ 1/2 = 4 r 0; 7/2 ÷ 1/2 = 7 r 0. This means 1/2 is the greatest common factor, the size of the cube we're looking for.
If the remainder is not zero, try again using the remainder from the division.
The volume of the prism is the volume of 28 cubes of size 1/2-inch. That volume is ...
28 × (1/2)(1/2)(1/2) = 28/8 = 7/2 . . . . cubic inches, as above.
