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?(1 point)
a. 8x^3+36x^2+54x+27
b. 8x^3-36x^2+54x+27
c. 8x^3+36x^2+54x-27
d. 8x^3-36x^2+54x-27
To find the new volume of the cube, we need to calculate the cube of the new side length. Let's use the binomial theorem for that.
The new side length is 2x reduced by 3 units, which gives us (2x - 3).
Using the binomial theorem, we can expand (2x - 3)^3 as:
(2x - 3)^3 = (2x)^3 - 3(2x)^2(3) + 3(2x)(3^2) - 3^3
Simplifying this expression:
8x^3 - 24x^2 + 36x - 27
Therefore, the correct expression for the new volume of the cube is:
8x^3 - 24x^2 + 36x - 27
Option a. 8x^3 + 36x^2 + 54x + 27 is incorrect.
Option b. 8x^3 - 36x^2 + 54x + 27 is incorrect.
Option c. 8x^3 + 36x^2 + 54x - 27 is incorrect.
Option d. 8x^3 - 36x^2 + 54x - 27 is correct.
To find the new volume of the cube, we need to apply the binomial theorem.
The binomial theorem states that for any two terms, (a+b)^n, the resulting expansion is given by:
(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
In this case, our terms are (2x - 3) and the power is 3, since we are cubing the expression.
Using the binomial theorem, the new volume of the cube is given by:
(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
Now let's expand this expression:
(2x - 3)^3 = C(3, 0)*(8x^3)*(1) + C(3, 1)*(4x^2)*(-3) + C(3, 2)*(2x)*9 + C(3, 3)*(1)*(-27)
Simplifying further, we get:
(2x - 3)^3 = 8x^3 - 36x^2 + 54x - 27
Therefore, the correct expression for the new volume of the cube is:
b. 8x^3 - 36x^2 + 54x - 27
To find the expression for the new volume of the cube, let's start by finding the original volume of the cube.
The original volume of the cube with side length 2x is given by (2x)^3. Expanding this using the binomial theorem we get:
(2x)^3 = 8x^3
Now, let's focus on the reduction of the side length of the cube by 3 units. This means the new side length is (2x - 3).
The new volume of the cube is given by (2x - 3)^3. Expanding this using the binomial theorem, we get:
(2x - 3)^3 = 8x^3 - 36x^2 + 54x - 27
Therefore, the correct expression for the new volume of the cube is:
8x^3 - 36x^2 + 54x - 27
So, the correct answer is option d. 8x^3 - 36x^2 + 54x - 27.