Final answer:
The half-life of the radioactive goo can be calculated using the decay formula and the given information. By solving the decay equation with the remaining goo after 75 minutes, we can determine the half-life. After finding the half-life, we can use the decay formula again to figure out how much goo will remain after 32 minutes.
Step-by-step explanation:
The half-life of a substance is the time required for half of the substance to decay. In the given experiment with radioactive goo, we are presented with an initial mass of 260 grams and a remaining mass of 8.125 grams after 75 minutes. To find the half-life, we will use the decay formula:
G(t) = G(0) × 0.5^(t/T)
Where G(t) is the final amount of substance, G(0) is the initial amount of substance, t is the time elapsed, and T is the half-life. Now, we need to solve for T:
8.125 = 260 × 0.5^(75/T)
Divide both sides by the initial amount:
8.125/260 = 0.5^(75/T)
Take the logarithm of both sides:
ln(8.125/260) = (75/T)×ln(0.5)
Solving for T gives us the half-life. To find how many grams of goo will remain after 32 minutes, substitute t = 32 into the equation G(t) with the determined value of T.
To answer the question specifically:
- The half-life T can be found using the decay equation.
- G(t) can be found by plugging the value of T and t into the decay formula.