What is the volume of a cube if its side length s is given by the expression 3x + 2y?

To find the volume of a cube, we use the formula:

Volume (V) = s³

where s is the side length of the cube.

In this case, the side length s is given as:

s = 3x + 2y

Now, to calculate the volume, we need to cube the side length:

V = (3x + 2y)³

To expand this expression, we can apply the binomial theorem:

According to the binomial theorem, (a + b)ⁿ = Σ (n choose k) * a^(n-k) * b^k.

Setting a = 3x, b = 2y, and n = 3, we have:

V = (3x + 2y)³ = (3x)³ + 3 * (3x)² * (2y) + 3 * (3x) * (2y)² + (2y)³

Now, let’s calculate each term:

• (3x)³ = 27x³
• 3 * (3x)² * (2y) = 3 * 9x² * 2y = 54x²y
• 3 * (3x) * (2y)² = 3 * (3x) * 4y² = 36xy²
• (2y)³ = 8y³

Putting it all together, we get:

V = 27x³ + 54x²y + 36xy² + 8y³

Thus, the volume of the cube with side length s = 3x + 2y is:

V = 27x³ + 54x²y + 36xy² + 8y³