At the beginning of an experiment, a scientist has 272 grams of radioactive goo. After 210 minutes, her sample has decayed to 4.25 grams.

(a)What is the half-life of the goo in minutes?

(b)Find a formula for G(t), the amount of goo remaining at time t.

(c)How many grams of goo will remain after 29 minutes?

Step 1 ：First, we need to understand the problem. The radioactive goo is decaying, which means its amount is decreasing over time. The half-life is the time it takes for half of the substance to decay. So we need to find out how long it takes for the goo to decrease from 272 grams to 136 grams.

Step 2 ：Let's denote the half-life as \(T\). We know that after 210 minutes, the goo has decayed to 4.25 grams. This means that the goo has halved \(\frac{210}{T}\) times.

Step 3 ：Since the goo started at 272 grams and ended at 4.25 grams, we can write the equation \(272 \times \left(\frac{1}{2}\right)^{\frac{210}{T}} = 4.25\).

Step 4 ：Solving this equation for \(T\), we get \(T = \frac{210}{\log_{\frac{1}{2}}\left(\frac{4.25}{272}\right)}\).

Step 5 ：Calculating the right side of the equation, we find that \(T = \boxed{30}\) minutes. So the half-life of the goo is 30 minutes.

Step 6 ：For part (b), we need to find a formula for \(G(t)\), the amount of goo remaining at time \(t\). Since the goo halves every 30 minutes, we can write \(G(t) = 272 \times \left(\frac{1}{2}\right)^{\frac{t}{30}}\).

Step 7 ：For part (c), we need to find out how many grams of goo will remain after 29 minutes. Substituting \(t = 29\) into the formula we found in part (b), we get \(G(29) = 272 \times \left(\frac{1}{2}\right)^{\frac{29}{30}}\).

Step 8 ：Calculating the right side of the equation, we find that \(G(29) = \boxed{138.6}\) grams. So after 29 minutes, there will be approximately 138.6 grams of goo remaining.