Final answer:
The domain of the vector-valued function r(t) is all real numbers t such that t is in the interval [-5, 5], which is determined by the condition that the square root component √(25 - t²) must be a real number.
Step-by-step explanation:
The question asks for the domain of the vector-valued function r(t) = √(25 - t²)i + t²j - 4tk. To determine the domain, we must look at the square root component √(25 - t²), as it restricts the possible values of t. The expression inside the square root, 25 - t², must be greater than or equal to zero to yield real number results, since we cannot have the square root of a negative number in real numbers.
Setting up the inequality 25 - t² ≥ 0 and solving for t, we find the allowable values for t, which are -5 ≤ t ≤ 5. Thus, the domain of the vector-valued function is all real numbers t such that t is in the interval [-5, 5].
The rest of the function, t²j and -4tk, does not further restrict the domain, as squaring a real number and multiplying by a real scalar can result in any real number, and do not impose any additional conditions on t.