A cube is shrunk so that its side lengths of 2x are reduced by 3 units. Using the Binomial Theorem, which of the following is the correct expression for the new volume of the cube?
8x3+36x2+54x+27
8 x cubed plus 36 x squared plus 54 x plus 27
8x3−36x2+54x+27
8 x cubed minus 36 x squared plus 54 x plus 27
8x3+36x2+54x−27
8 x cubed plus 36 x squared plus 54 x minus 27
8x3−36x2+54x−27
The correct expression for the new volume of the cube is:
8x^3 - 36x^2 + 54x - 27
This is because when the side length of 2x is reduced by 3 units, the new side length becomes 2x - 3. The volume of a cube is given by (side length)^3, so the new volume can be expressed as (2x - 3)^3. Applying the Binomial Theorem to expand this expression, we get:
(2x - 3)^3 = 8x^3 - 36x^2 + 54x - 27
To find the expression for the new volume of the cube after shrinking, we need to use the formula for the volume of a cube, which is (side length)^3.
The original side length of the cube is 2x, and it is reduced by 3 units. Therefore, the new side length of the cube will be (2x - 3).
To find the expression for the new volume, we substitute (2x - 3) into the formula for the volume of a cube:
Volume = (2x - 3)^3
Using the Binomial Theorem, we can expand this expression:
Volume = (2x)^3 - 3(2x)^2(3) + 3(2x)(3)^2 - (3)^3
Simplifying this expression gives us:
Volume = 8x^3 - 36x^2 + 54x - 27
Therefore, the correct expression for the new volume of the cube is:
8x^3 - 36x^2 + 54x - 27
To find the expression for the new volume of the cube after it is shrunk, we need to consider the change in the side lengths.
The original side length of the cube is 2x. When it is reduced by 3 units, the new side length becomes (2x - 3).
The volume of a cube is given by V = side length^3. So, the original volume of the cube is (2x)^3 = 8x^3.
Now, let's use the Binomial Theorem to expand the expression (2x - 3)^3.
The Binomial Theorem states that (a + b)^n = C(n, 0) * a^n * b^0 + C(n, 1) * a^(n-1) * b^1 + C(n, 2) * a^(n-2) * b^2 + ... + C(n, n-1) * a^1 * b^(n-1) + C(n, n) * a^0 * b^n,
where C(n, k) represents the binomial coefficient and is calculated as n! / (k! * (n-k)!).
In our case, (a + b) = (2x - 3), and n = 3.
Now, let's expand (2x - 3)^3 using the Binomial Theorem:
(2x - 3)^3
= C(3, 0) * (2x)^3 * (-3)^0 + C(3, 1) * (2x)^2 * (-3)^1 + C(3, 2) * (2x)^1 * (-3)^2 + C(3, 3) * (2x)^0 * (-3)^3
= 1 * 8x^3 * 1 + 3 * 4x^2 * -3 + 3 * 2x * 9 + 1 * 1 * -27
= 8x^3 - 36x^2 + 54x - 27
Therefore, the correct expression for the new volume of the cube is 8x^3 - 36x^2 + 54x - 27.