A rectangular prism with a volume of 666 cubic units is filled with cubes with side lengths of \dfrac12 2 1 ? start fraction, 1, divided by, 2, end fraction unit. How many \dfrac12 2 1 ? start fraction, 1, divided by, 2, end fraction unit cubes does it take to fill the prism?
Accepted Solution
A:
Answer:48 cubes.Step-by-step explanation:Let n represent number of cubes with each side [tex]\frac{1}{2}[/tex] units.We have been given that a rectangular prism with a volume of 6 cubic units is filled with cubes with side lengths of [tex]\frac{1}{2}[/tex] units. We are asked to find the number of cubes that will fill the prism.We know that volume of n cubes will be equal to volume of rectangular prism.[tex]n\cdot \text{Volume of each cube}=\text{Volume of rectangular prism}[/tex]We also know that volume of a cube with side length 'x' is equal to [tex]x^3[/tex].[tex]\text{Volume of each cube}=(\frac{1}{2}\text{ Unit})^3[/tex][tex]\text{Volume of each cube}=\frac{1^3}{2^3}\text{ Unit}^3[/tex][tex]\text{Volume of each cube}=\frac{1}{8}\text{ Unit}^3[/tex][tex]n\cdot \frac{1}{8}\text{ Unit}^3=6\text{ Unit}^3[/tex][tex]n\cdot \frac{1}{8}=6[/tex][tex]n\cdot \frac{1}{8}\cdot 8=6\cdot 8[/tex][tex]n=48[/tex]Therefore, it will take 48 cubes with each side [tex]\frac{1}{2}[/tex] units to fill the prism.