3. A desintegração de certa substância é descrita pela lei \( M(t)=256(4)^{-0,125 t} \), em que 256 g é a massa inicial dessa substância, \( t \) representa o tempo, em horas e \( M(t) \) a massa, em gramas dessa substância no instante \( t \). Quanto tempo leva para que uma amostra seja reduzida a 32 g .

The formula \( A=24.2 e^{0.0766 t} \) models the population of a US state, \( A \), in millions, \( t \) years after 2000 . a. What was the fopulation of the state in 2000 ? b. When will the population of the state reach 31.4 million? a. In 2000 , the population of the state was \( \square \) million. b. The population of the state will reach 31.4 million in the year \( \square \). (Round to the nearest year as needed.)

Use the given functions to answer the question. \( \begin{array}{l}f(x)=2^{x} \\ g(x)=x-3 \\ \text { What is the domain of } \frac{f(x)}{g(x)}=\frac{2^{x}}{x-3} \text { ? } \\ \text { A. the set of all real numbers } \\ \text { B. the set of all real numbers greater than } 0 \\ \text { D. the set of all real numbers not equal to } 0 \\ \text { E. the set of all real numbers not equal to } 3\end{array} \) B. thers greater than 3

\( \frac{\log x-\sqrt{x^{2}-3 x+2}}{\sqrt{x^{2}-4}} \) For Domain

Find \( \|\mathbf{v}\| \). \( \begin{array}{l}\mathbf{v}=9 \mathbf{i}-12 \mathbf{j} \\ \|\mathbf{v}\|=\square \text { (Type an exact answer, using radicals as needed. Simplify your answer.) }\end{array} \)

Write the vector v in the form \( \mathrm{ai}+\mathrm{bj} \), given its magnitude \( \|\mathrm{v}\| \) and the angle \( \alpha \) it makes with the positive \( x \)-axis, \[ \alpha v=45^{\circ} \] \( \mathrm{v}=\square \) (Simplify your answer. Type an exact answer using radicals as needed. Type your answer in the form ai + bj.)