To understand the problem better, let's break it down step by step.
1. Understanding the Volume Function v(x):
- The volume function [tex]\( v(x) \)[/tex] is given as:
[tex]\[
v(x) = x^3 - 12x^2 + 35x
\][/tex]
2. Interpreting the Function:
- Here, [tex]\( x \)[/tex] represents the side length of the cubic case in inches.
- The given function [tex]\( v(x) \)[/tex] represents the volume of a rectangular prism that is related to the size of the cube.
3. Analyzing the Function:
- The term [tex]\( x^3 \)[/tex] represents the volume of the cube since the volume of a cube with side length [tex]\( x \)[/tex] is [tex]\( x \times x \times x \)[/tex].
- The terms [tex]\( -12x^2 \)[/tex] and [tex]\( +35x \)[/tex] represent modifications to this cubic volume to model the rectangular prism, taking into account some changes such as reduction or addition in dimensions.
4. Solution Presentation:
The volume function for the rectangular prism is:
[tex]\[
v(x) = x^3 - 12x^2 + 35x
\][/tex]
This function correctly represents the volume of the redesigned case as described.
