Question

Rosa has 16 unit cubes. How many different rectangular prisms can she build with the cubes?

Answer 1

Hint: Here in this question, we have to find the number of different ways of rectangular prisms formed by using 16 unit cubes. This can be solved by using the factors of 16 or using the volume of the prism and cube; we get the required possibilities to build the rectangular prism using the unit cubes.

Complete step by step solution:
A rectangular prism is a three-dimensional shape, having six faces, and all the faces of the prism are rectangles. Both the base and the top of a rectangular prism must be rectangles.
Since the rectangular prism is formed by the unit cubes, each dimension of the prism must be a whole number of cubes.
Given that Rosa has 16 unit cubes, we need to find the total number of different combinations of three whole numbers that have a product of 16. Cubes in a rectangular prism depend on the dimensions of the prism, and how large of cubes are. As we know, The volume of the prism = volume of cube
\[
\Rightarrow l \times b \times h = 16\left( {1 \times 1 \times 1} \right)
\],
since it is a unit cube
\[
\Rightarrow l \times b \times h = 16
\]
 number of possibilities for 16 are \[
\Rightarrow 16 \times 1 \times 1
\]

\[
\Rightarrow 1 \times 2 \times 8
\]
                                                                                                                                                                                 \[
\Rightarrow 1 \times 4 \times 4
\]

\[ \Rightarrow 2 \times 2 \times 4
\]
 do not take fractions into consideration because it is a unit cube only integer values will be accepted for \[l\],\[b\]and \[h\].
Hence, Rosa can build 4 different rectangular prisms with 16 unit cubes.
So, the correct answer is “4”.

Note: This is a word problem. As we know the relation between the prism and cube is the volume of the prism is equal to the volume of the cube. While simplifying we use the simple arithmetic operations namely, addition, subtraction, multiplication and division. We must know the table of multiplication.