Find the volume of the right rectangular prism using the cube method Each cube is 1/3 inch on each side What is the volume of the rectangular prism? Provide your answer as a fraction reduced to the lowest term
If each cube is 1/3 inch on each side, then the height, length, and width of the rectangular prism must be divisible by 1/3. Let's call the height h, length l, and width w.
Using the cube method, we can find the volume of the rectangular prism by counting the number of cubes that fit inside.
First, let's fill up one layer of the rectangular prism:
[asy]
import three;
defaultpen(linewidth(0.7));
draw(unitcube);
draw(shift((1,0,0))*unitcube);
draw(shift((2,0,0))*unitcube);
draw(shift((0,1,0))*unitcube);
draw(shift((1,1,0))*unitcube);
draw(shift((2,1,0))*unitcube);
draw(shift((0,2,0))*unitcube);
draw(shift((1,2,0))*unitcube);
draw(shift((2,2,0))*unitcube);
draw(shift((0,0,0))*unitcube);
label("$w$",(2,-0.5,0),N);
label("$l$",(0.5,2.5,0),N);
label("$1/3$",(0,-0.2,0),W);
label("$1/3$",(1,-0.2,0),W);
label("$1/3$",(2,-0.2,0),W);
label("$1/3$",(4/3,1.8,0),N);
label("$1/3$",(5/3,1.8,0),N);
label("$1/3$",(2,1.8,0),N);
label("$1/3$",(1/3,1/3,-0.2),S);
label("$1/3$",(1/3,4/3,-0.2),S);
label("$1/3$",(1/3,2,-0.2),S);
[/asy]
The length of this layer is l, the width is w, and the height is 1/3. To find the number of cubes in this layer, we can divide the length by 1/3, the width by 1/3, and the height by 1/3:
number of cubes = (l / (1/3)) * (w / (1/3)) * (1/3) = 3lw
Now we need to find the number of layers. The height of the rectangular prism is h, and each layer has a height of 1/3. So the number of layers is:
number of layers = h / (1/3) = 3h
Therefore, the total number of cubes in the rectangular prism is:
total number of cubes = number of cubes in one layer * number of layers = 3lw * 3h = 9lwh
Finally, we can find the volume of the rectangular prism by multiplying the total number of cubes by the volume of each cube:
volume of rectangular prism = total number of cubes * volume of each cube = 9lwh * (1/3)^3 = lwh/3
So the volume of the right rectangular prism is lwh/3, which is a fraction reduced to the lowest term.