(a) \( 3+\frac{3}{4}+\frac{3}{16}+\frac{3}{64}+\ldots+\frac{3}{4^{k}}+\ldots \) is (b) \( 25-5+1-\frac{1}{5}+\ldots+\frac{(-1)^{k} 25}{5^{k}}+\ldots \) divergent (c) \( 4+9+\frac{81}{4}+\frac{729}{16}+\ldots+4\left(\frac{9}{4}\right)^{k}+. \quad \) convergent

\( \int \cosh ^ { 2 } ( x - 3 ) \sinh ( x - 3 ) d x \)

At the beginning of an experiment, a scientist has 260 grams of radioactive goo. After 210 minutes, her sample has decayed to 8.125 grams. What is the half-life of the goo in minutes? 42 Find a formula for \( G(t) \), the amount of goo remaining at time \( t \). \( G(t)=260 e^{-0.0165 t} \) How many grams of goo will remain after 47 minutes? You may enter the exact value or round to 2 decimal places.

You go to the doctor and he gives you 12 milligrams of radioactive dye. After 24 minutes, 4.75 milligrams of dye remain in your system. To leave the doctor's office, you must pass through a radiation detector without sounding the alarm. If the detector will sound the alarm if more than 2 milligrams of the dye are in your system, how long will your visit to the doctor take, assuming you were given the dye as soon as you arrived? Give your answer to the nearest minute. You will spend minutes at the doctor's office.

b) \( \sum_{n=1}^{\infty} \sin \frac{\pi}{n^{2}} \)

\( \lim _ { n \rightarrow \infty } [ \frac { 1 } { \sqrt { 4 n ^ { 2 } - 1 } } + \frac { 1 } { \sqrt { 4 n ^ { 2 } - 2 ^ { 2 } } } + \cdots + \frac { 1 } { \sqrt { 3 n ^ { 2 } } } ] \)