Find the domain of the following vector-valued function. \( r(t)=\sqrt{81-t^{2}} i+\sqrt{t} j+\frac{8}{\sqrt{6+t}} k \) Select the correct choice below and fill in any answer boxes within your choice. \( \begin{array}{ll}\text { A. }\{t: t \leq \square \text { or } t>\square\} & \text { B. }\{t: \square<\square \text { or } t \geq \square\} \\ \text { E. }\{t: t<\square \text { or } t>\square\} & \text { D. }\{t: t \leq \square \text { or } t \geq \square\} \\ \text { G. }\{t: \square \leq t<\square\} & \text { F. }\{t: \square

Ask by Reid Sanders. in the United States

Feb 03,2025

To find the domain of the vector-valued function \( r(t) \), we need to consider the restrictions imposed by each component of the function: 1. The term \( \sqrt{81 - t^2} \) requires that \( 81 - t^2 \geq 0 \), which leads to \( -9 \leq t \leq 9 \). 2. The term \( \sqrt{t} \) requires that \( t \geq 0 \). 3. The term \( \frac{8}{\sqrt{6 + t}} \) imposes that \( 6 + t > 0 \) or \( t > -6 \) (but since \( t \geq 0 \) is a stricter condition, we prioritize that). By combining these conditions, \( t \) must be in the range \( [0, 9] \). Thus, in the correct choice format: \( \begin{array}{ll}\text { G. }\{t: 0 \leq t < 9\} & \end{array} \) \( \begin{array}{ll}\text { E }\{0\}\end{array} \)