First, we need to convert the mixed numbers into fractions
[tex]3\frac{1}{3}=\frac{3\cdot3+1}{3}=\frac{10}{3}\text{ ft}[/tex][tex]1\frac{1}{3}=\frac{1\cdot3+1}{3}=\frac{4}{3}\text{ ft}[/tex][tex]2\frac{1}{3}=\frac{2\cdot3+1}{3}=\frac{7}{3}\text{ ft}[/tex]To find how many cubes fit on the length, we need to divide 10/3 by 1/3, as follows:
[tex]\frac{\frac{10}{3}}{\frac{1}{3}}=\frac{10}{3}\cdot3=10[/tex]To find how many cubes fit on the width, we need to divide 4/3 by 1/3, as follows:
[tex]\frac{\frac{4}{3}}{\frac{1}{3}}=\frac{4}{3}\cdot3=4[/tex]To find how many cubes fit on the height, we need to divide 7/3 by 1/3, as follows:
[tex]\frac{\frac{7}{3}}{\frac{1}{3}}=\frac{7}{3}\cdot3=7[/tex]Then, the length has 10 cubes, the width has 4 cubes, and the height has 7 cubes.
The volume of each cube is:
[tex]V=a^3=(\frac{1}{3})^3=\frac{1}{27}ft^3[/tex]The number of cubes that fit in the rectangular prism is: 10x4x7 = 280. Therefore, the volume of the prism is
[tex]280\cdot\frac{1}{27}=\frac{270+10}{27}=\frac{270}{27}+\frac{10}{27}=10\frac{10}{27}ft^3[/tex]