The volume of a rectangular prism whose dimensions are binomials with integer coefficients is modeled by the function V(x) = x^3 − 14x^2 + 49x − 36. Given that x − 1 and x − 9 are two of the dimensions, find the missing dimension of the prism.

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Adetunmbiadekunle Adetunmbiadekunle · Answer 1 · 2020-12-09T01:32:53+01:00

Answer:

The missing dimension of the prism is (x-4)

Step-by-step explanation:

Here, we want to find the missing dimensions of the prism

To get the volume, we are to multiply three dimensions since we are talking about volume

Mathematically, to find the third dimension, we need to divide the original polynomial by the product of the two other dimensions

The product of the two other dimensions is;

(x-1)(x-9) = x^2-9x-x + 9

= x^2-10x+ 9

So we divide;

(x^3-14x^2+49x-36)/(x^2-10x+ 9)

We can use long division to get this

using long division, the answer here is x-4

MrRoyal MrRoyal · Answer 2 · 2021-12-09T10:30:35+01:00

The volume of a prism is the product of its dimensions.

The missing dimension of the prism is x - 4

The volume function is given as:

[tex]\mathbf{V(x) = x^3 - 14x^2 + 49x - 36}[/tex]

Let the missing dimension of the prism be y.

So, the volume is calculated using:

[tex]\mathbf{V(x) = y \times (x - 9) \times (x -1)}[/tex]

Multiply the factors

[tex]\mathbf{V(x) = y \times ( x^2-9x-x + 9)}[/tex]

[tex]\mathbf{V(x) = y \times ( x^2-10x + 9)}[/tex]

Equate both expressions of volume

[tex]\mathbf{y \times ( x^2-10x + 9) = x^3 - 14x^2 + 49x - 36}[/tex]

Make y the subject

[tex]\mathbf{y = \frac{x^3 - 14x^2 + 49x - 36}{x^2-10x + 9 }}[/tex]

Factor the numerator

[tex]\mathbf{y = \frac{(x - 4)(x^2-10x + 9 )}{x^2-10x + 9 }}[/tex]

Cancel out common factors

[tex]\mathbf{y = x - 4}[/tex]

Hence, the missing dimension of the prism is x - 4