Question

Two night rectangular prisms are shown. If each prism is packed with cubes with the side length of kin, which prism has the most cubes? Prism 2. Prism 1. Ain Alw N in 27 3 in. 1 3 in. 2. 12 in 3 1ā in. Slup 1. Find the number of the smallcubes with the fractional edge side to fill the prismi (divide the prism edges lengths by the length of the side edge le find out how many cubes fit in) O Volume of Prism 1 is greater by 120 cubes Volume of Prism 2 is greater by 120 cubes

Answer 1

In order to calculate the number of small boxes that fit in each prism, we can calculate the volume of each prism and then divide them by the volume of the small cubes, like this:

• Prism 1

$\begin{gathered} V=4\times1\frac{1}{2}\times2\frac{3}{4} \\ V=4\times\frac{1\times2+1}{2}\times\frac{2\times4+3}{4} \\ V=4\times\frac{3}{2}\times\frac{11}{4} \\ V=\frac{4\times3\times11}{2\times4}=\frac{132}{8}=16.5 \end{gathered}$

• Prism 2

$\begin{gathered} V=3\times1\frac{3}{4}\times3\frac{1}{2} \\ V=3\times\frac{1\times4+3}{4}\times\frac{3\times2+1}{2} \\ V=3\times\frac{4+3}{4}\times\frac{6+1}{2} \\ V=3\times\frac{7}{4}\times\frac{7}{2} \\ V=\frac{3\times7\times7}{4\times2}=\frac{196}{8}=24.5 \end{gathered}$

• Cube

$V=\frac{1}{4}\times\frac{1}{4}\times\frac{1}{4}=\frac{1}{4^3}=\frac{1}{64}$

As mentioned, by dividing the volume of each prism by the volume of the cube, we get the number of cubes that fit in each figure:

-For the first prism:

$n1=\frac{16.5}{\frac{1}{64}}=1056$

-For the second prism:

$\frac{24.5}{\frac{1}{64}}=1568$

As you can see, more cubes fit in the second prism, then the correct answer is the second one.