Question

the side length, s, of a cube is 3x + 2y. if v = s³, what is the volume of the cube?
27x³ + 18x²y + 12xy² + 2y³
27x³ + 54x²y + 36xy² + 8y³
27x³ + 54x²y + 18xy² + 2y³
3x³ + 18x²y + 36xy² + 8y³

Explanation:

Step1: Recall the formula for cube volume

The volume \( V \) of a cube with side length \( s \) is given by \( V = s^3 \). Here, \( s = 3x + 2y \), so we need to expand \( (3x + 2y)^3 \).

Step2: Use the binomial expansion formula \((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\)

Let \( a = 3x \) and \( b = 2y \). Then:

• \( a^3 = (3x)^3 = 27x^3 \)
• \( 3a^2b = 3(3x)^2(2y) = 3(9x^2)(2y) = 54x^2y \)
• \( 3ab^2 = 3(3x)(2y)^2 = 3(3x)(4y^2) = 36xy^2 \)
• \( b^3 = (2y)^3 = 8y^3 \)

Step3: Combine the terms

Adding these terms together: \( 27x^3 + 54x^2y + 36xy^2 + 8y^3 \)

Answer:

\( 27x^3 + 54x^2y + 36xy^2 + 8y^3 \) (the second option in the list)