Certainly! Let's tackle this step by step:
### (a) Writing the Volume Function
The volume [tex]\( V \)[/tex] of the box is given as a function of [tex]\( x \)[/tex]. The volume function is:
[tex]\[ V(x) = 4x^3 - 144x^2 + 1296x \][/tex]
### (b) Determining the Domain of the Function
To determine the domain of the function [tex]\( V(x) \)[/tex], we need to consider the conditions under which the volume is non-negative since the volume of a physical box must be zero or positive.
This means we need to solve the inequality:
[tex]\[ 4x^3 - 144x^2 + 1296x \geq 0 \][/tex]
The solutions to this inequality will give us the range of [tex]\( x \)[/tex] for which the volume function [tex]\( V(x) \)[/tex] is valid.
Upon solving the inequality, we find that:
[tex]\[ 0 \leq x < \infty \][/tex]
### Summary
- The volume function is:
[tex]\[ V(x) = 4x^3 - 144x^2 + 1296x \][/tex]
- The domain of the function [tex]\( V(x) \)[/tex] is:
[tex]\[ [0, \infty) \][/tex]
In interval notation, the domain of [tex]\( V \)[/tex] is:
[tex]\[ [0, \infty) \][/tex]
This means [tex]\( x \)[/tex] can take any value starting from 0 and extending to positive infinity.
