Find the domain of $y=4 \sqrt{x-8}$.
The function given is \(y = 4\sqrt{x - 8}\). To determine the domain, we must consider the values of \(x\) for which the expression is defined within the set of real numbers. A square root function, \(\sqrt{f(x)}\), is defined if and only if the expression inside the radical (the radicand) is non-negative, meaning \(f(x) \geq 0\).
For the function \(y = 4\sqrt{x - 8}\), the expression under the square root is \(x - 8\). Applying the requirement for the square root to be defined, we set up the inequality:
\[x - 8 \geq 0\]
To solve for \(x\), we add 8 to both sides of the inequality:
\[x \geq 8\]
This means the function is defined for all real values of \(x\) that are greater than or equal to 8.
We have found that the domain of the function is \(x \geq 8\). Comparing this result with the given options, we can see that it corresponds to the final option provided:
Thus, the correct choice is \(x \geq 8\).
In practical applications, square root functions often represent relationships where the input cannot be negative, such as the time elapsed since an event or a starting position. For instance, if a company's production cost model is represented by \(C(t) = 4\sqrt{t - 8}\), where \(t\) represents the number of days after a project starts, the domain \(t \geq 8\) signifies that the model only becomes applicable 8 days after the initiation of the project. Determining the domain helps in understanding the boundaries and operational limits of such real-world mathematical models.