Question

Alex has 64 cubes, with the dimensions in ft (1/2) he uses all the cubes to fill a box shaped like a larger rectangular prism. there are no gaps between the cubes.

A. what is the volume, in cubic feet, of the larger rectangular prism?

B. what is the possible set of dimensions , in feet, of the larger rectangular prism?

Answer 1

we know that

the dimensions of the cube are------> 1/2 ft x 1/2 ft x 1/2 ft

Part A)
[the volume of the larger rectangular prism]=64*[(1/2)*(1/2)*(1/2)]=8 ft³

the answer Part A) is
8 ft³

Part B) B. what is the possible set of dimensions , in feet, of the larger rectangular prism?

we know that
[volume of rectangular prism]=L*W*H
volume of 1 cube=1/8 ft³
volume of 64 cubes=8 ft³
so

the answer Part B) is
any combination of whole cubes such that
L*W*H=64 cubes

Examples:
4 x 4 x 4-------> 4 cubes x 4 cubes x 4 cubes-----> 64 cubes
4 cubes-----> 4*(1/2)----> 2 ft
so
2 ft x 2 ft x 2 ft

8 x 8 x 1-------> 8 cubes x 8 cubes x 1 cube-----> 64 cubes
8 cubes-----> 8*(1/2)-----> 4 ft
1 cube-------> 1*(1/2)-----1/2 ft
so
4 ft x 4 ft x1/2 ft

8 x 4 x 2------> 8 cubes x 4 cubes x 2 cubes-----> 64 cubes
8 cubes-------> 8*(1/2)------> 4 ft
4 cubes-------> 4*(1/2)------> 2 ft
2 cubes-------> 2*(1/2)-----> 1 ft
so
4 ft x 2 ft x 1 ft