The side length, [tex]$s$[/tex], of a cube is [tex]$3 x+2 y$[/tex]. If [tex]$V=s^3$[/tex], what is the volume of the cube?
A. [tex]$3 x^3+18 x^2 y+36 x y^2+8 y^3$[/tex]
B. [tex]$27 x^3+54 x^2 y+18 x y^2+2 y^3$[/tex]
C. [tex]$27 x^3+18 x^2 y+12 x y^2+2 y^3$[/tex]
D. [tex]$27 x^3+54 x^2 y+36 x y^2+8 y^3$[/tex]
Answer (2)
Substitute the side length s = 3 x + 2 y into the volume formula V = s 3 .
Expand the expression ( 3 x + 2 y ) 3 using the binomial theorem.
Simplify each term: ( 3 x ) 3 = 27 x 3 , 3 ( 3 x ) 2 ( 2 y ) = 54 x 2 y , 3 ( 3 x ) ( 2 y ) 2 = 36 x y 2 , ( 2 y ) 3 = 8 y 3 .
Combine the terms to get the volume: V = 27 x 3 + 54 x 2 y + 36 x y 2 + 8 y 3 . The volume of the cube is 27 x 3 + 54 x 2 y + 36 x y 2 + 8 y 3 .
Explanation
Understanding the Problem We are given that the side length of a cube is s = 3 x + 2 y and the volume is V = s 3 . We need to find the volume of the cube in terms of x and y .
Substituting the Side Length Substitute s = 3 x + 2 y into the volume formula V = s 3 to get V = ( 3 x + 2 y ) 3 .
Expanding the Expression Expand the expression ( 3 x + 2 y ) 3 using the binomial theorem or by direct multiplication. The binomial expansion is ( a + b ) 3 = a 3 + 3 a 2 b + 3 a b 2 + b 3 . In our case, a = 3 x and b = 2 y .
Applying the Binomial Theorem So, V = ( 3 x ) 3 + 3 ( 3 x ) 2 ( 2 y ) + 3 ( 3 x ) ( 2 y ) 2 + ( 2 y ) 3 .
Simplifying Simplify the expression to find the volume V .
Calculating the First Term Calculate ( 3 x ) 3 = 27 x 3 .
Calculating the Second Term Calculate 3 ( 3 x ) 2 ( 2 y ) = 3 ( 9 x 2 ) ( 2 y ) = 54 x 2 y .
Calculating the Third Term Calculate 3 ( 3 x ) ( 2 y ) 2 = 3 ( 3 x ) ( 4 y 2 ) = 36 x y 2 .
Calculating the Fourth Term Calculate ( 2 y ) 3 = 8 y 3 .
Final Volume Therefore, V = 27 x 3 + 54 x 2 y + 36 x y 2 + 8 y 3 .
Examples
Cubes are fundamental shapes in geometry and appear in many real-world applications. For example, calculating the volume of a cubic storage container helps determine how much it can hold. In architecture, understanding the volume of cubic spaces is crucial for ventilation and heating calculations. Moreover, in manufacturing, cubic blocks are often used as building components, and knowing their volume is essential for material estimation and cost analysis. The formula V = s 3 , where s is the side length, allows us to quickly determine the volume, which is vital for planning and resource management.
The volume of the cube with side length s = 3 x + 2 y is 27 x 3 + 54 x 2 y + 36 x y 2 + 8 y 3 . This matches option D in the list of answers. Thus, the correct answer is option D.
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