A rectangular prism has a volume of
4x(cubed) + 30 x(squared) +36x. What linear expressions can represent possible dimensions of the prism?
Please explain.
The volume factors into four linear factors:
2x(x+6)(2x+3)
Since we need three values for the dimensions of the prism, so there are
C(4,2)=6 sets of dimensions, i.e. choose 2 factors out of 4 to reduce to one dimension. The two other dimensions will be used as is.
Example dimensions are:
2x (x+6) (2x+3)
2 x(x+6) (2x+3)
x 2(x+6) (2x+3)
...
up to 6 sets.
To find the possible dimensions of the rectangular prism with a given volume, we need to factorize the volume expression and look for linear factors.
The given volume expression is: 4x^3 + 30x^2 + 36x.
First, we can factor out the greatest common factor, which is 2x, giving us:
2x(2x^2 + 15x + 18).
Next, we need to factor the quadratic expression inside the parentheses: 2x^2 + 15x + 18.
To factor this quadratic expression, we need to find two numbers that multiply to give 18 and add up to 15. The numbers that satisfy these conditions are 3 and 12 (since 3 * 12 = 36 and 3 + 12 = 15).
So, we can rewrite the quadratic expression as:
2x(x + 3)(2x + 6).
Now, we have factored the entire volume expression as:
2x(x + 3)(2x + 6).
From this factorization, we can see that the possible dimensions of the rectangular prism are:
Length: 2x or x + 3 (since 2x and x + 3 are linear expressions)
Width: x or x + 6 (since x and 2x + 6 are linear expressions)
Height: x + 3 or 2x + 6 (since x + 3 and 2x + 6 are linear expressions)
Therefore, the linear expressions that can represent the possible dimensions of the rectangular prism are 2x, x + 3, x, and 2x + 6.
To determine the linear expressions that can represent possible dimensions of the rectangular prism, we need to find the factors of the given volume expression: 4x^3 + 30x^2 + 36x.
First, let's factor out the common factor of 2x from each term:
2x(2x^2 + 15x + 18).
Now, we need to factor the quadratic expression inside the parentheses: 2x^2 + 15x + 18.
We can use the factoring method or the quadratic formula to find the factors. Let's use factoring:
(2x + 3)(x + 6).
Finally, the factored form of the volume expression is 2x(2x + 3)(x + 6).
Now, let's identify the possible dimensions of the rectangular prism based on these factors. The length, width, and height of the rectangular prism can be represented by the factors of the volume expression. In this case, the possible dimensions are:
Length: 2x + 3
Width: x + 6
Height: 2x
So, the linear expressions that can represent the possible dimensions of the rectangular prism are 2x + 3, x + 6, and 2x.