To solve for the value of [tex]\( x \)[/tex] and determine the length of the longest side of the rectangular prism, we need to follow these steps:
1. Set up the volume equation: Given the dimensions of the prism [tex]\( (x+1) \)[/tex] cm, [tex]\( (2x) \)[/tex] cm, and [tex]\( (x+6) \)[/tex] cm, we should start by setting up the volume equation. The volume [tex]\( V \)[/tex] is given by the product of these dimensions:
[tex]\[
V = (x + 1)(2x)(x + 6)
\][/tex]
Given that the volume is 400 cubic centimeters, we write:
[tex]\[
(x + 1)(2x)(x + 6) = 400
\][/tex]
2. Expand and simplify: First, expand the left-hand side of the equation:
[tex]\[
(x + 1)(2x)(x + 6) = 2x(x^2 + 7x + 6)
\][/tex]
[tex]\[
2x(x^2 + 7x + 6) = 2x^3 + 14x^2 + 12x
\][/tex]
Thus, the equation becomes:
[tex]\[
2x^3 + 14x^2 + 12x = 400
\][/tex]
3. Set the equation to zero: Move all terms to one side of the equation to set it to zero:
[tex]\[
2x^3 + 14x^2 + 12x - 400 = 0
\][/tex]
4. Solve the cubic equation: Use a graphing calculator or other appropriate numerical methods to find the roots of the equation [tex]\( 2x^3 + 14x^2 + 12x - 400 = 0 \)[/tex].
Using a graphing calculator:
- Plot the function [tex]\( f(x) = 2x^3 + 14x^2 + 12x - 400 \)[/tex].
- Identify the x-intercepts (roots) of this function.
After plotting, we find that one of the real roots is approximately [tex]\( x = 4 \)[/tex].
5. Calculate the dimensions: Substitute [tex]\( x = 4 \)[/tex] back into the original dimensions to find each side:
[tex]\[
\text{Side 1} = x + 1 = 4 + 1 = 5 \text{ cm}
\][/tex]
[tex]\[
\text{Side 2} = 2x = 2(4) = 8 \text{ cm}
\][/tex]
[tex]\[
\text{Side 3} = x + 6 = 4 + 6 = 10 \text{ cm}
\][/tex]
6. Determine the longest side: Compare the calculated dimensions to find the longest side:
[tex]\[
\text{Side 1} = 5 \text{ cm}, \quad \text{Side 2} = 8 \text{ cm}, \quad \text{Side 3} = 10 \text{ cm}
\][/tex]
The longest side is [tex]\( \text{Side 3} = 10 \text{ cm} \)[/tex].
Therefore, the length of the longest side is [tex]\(\boxed{10 \text{ centimeters}}\)[/tex].
