The side length, [tex]$s$[/tex], of a cube is [tex]$x-2 y$[/tex]. If [tex]$V=s^3$[/tex], what is the volume of the cube?
A. [tex]$x^3-6 x^2 y+12 x y^2-8 y^3$[/tex]
B. [tex]$x^3+6 x^2 y+12 x y^2+8 y^3$[/tex]
C. [tex]$3 x^3-6 x^2 y+12 x y^2-24 y^3$[/tex]
D. [tex]$3 x^3+6 x^2 y+12 x y^2+24 y^3$[/tex]
Answer (1)
Substitute the side length s = x − 2 y into the volume formula V = s 3 .
Expand the expression ( x − 2 y ) 3 using direct multiplication.
Combine like terms to simplify the expression.
The volume of the cube is V = x 3 − 6 x 2 y + 12 x y 2 − 8 y 3 .
Explanation
Problem Setup We are given that the side length of a cube is s = x − 2 y and the volume of the cube is V = s 3 . Our goal is to find the volume V in terms of x and y .
Substitution To find the volume, we substitute the expression for s into the volume formula: V = ( x − 2 y ) 3
Expanding the Expression - Step 1 Now we need to expand the expression ( x − 2 y ) 3 . We can use the binomial theorem or direct multiplication. Let's use direct multiplication: ( x − 2 y ) 3 = ( x − 2 y ) ( x − 2 y ) ( x − 2 y ) First, we multiply ( x − 2 y ) ( x − 2 y ) : ( x − 2 y ) ( x − 2 y ) = x 2 − 2 x y − 2 x y + 4 y 2 = x 2 − 4 x y + 4 y 2
Expanding the Expression - Step 2 Next, we multiply the result by ( x − 2 y ) : ( x 2 − 4 x y + 4 y 2 ) ( x − 2 y ) = x 3 − 4 x 2 y + 4 x y 2 − 2 x 2 y + 8 x y 2 − 8 y 3
Combining Like Terms Now, we combine like terms: x 3 − 4 x 2 y − 2 x 2 y + 4 x y 2 + 8 x y 2 − 8 y 3 = x 3 − 6 x 2 y + 12 x y 2 − 8 y 3
Final Volume Therefore, the volume of the cube is: V = x 3 − 6 x 2 y + 12 x y 2 − 8 y 3
Examples
Cubes are fundamental geometric shapes that appear in various real-world applications. For instance, calculating the volume of a cubic storage container helps determine its capacity. In architecture, understanding the volume of cubic building blocks is essential for structural design and material estimation. Moreover, in chemistry, the volume of a crystal with a cubic lattice structure is crucial for determining its density and other physical properties. The expansion of ( x − 2 y ) 3 can be seen as a way to model how the volume of a cube changes when its side length is altered, which is useful in engineering and design.
