Problem 18 - College Algebra

Question

The half-life of the radioactive element plutonium-239 is \(25,000\) years. If 16 grams of plutonium- 239 are initially present, how many grams are present after \(25,000\) years? \(50,000\) years? \(75,000\) years? \(100,000\) years? \(125,000\) years?

Step-by-Step Solution

Answer

After 25,000 years there are 8 grams left. After 50,000 years there are 4 grams left. After 75,000 years there are 2 grams left. After 100,000 years there is 1 gram left. After 125,000 years there is 0.5 grams left.

Step 1: Calculating the amount after 25,000 years

For the first 25,000 years, half of the initial 16 grams of plutonium-239 would have decayed, leaving us with half. Therefore, the final amount is \(16/2 = 8\) g.

Step 2: Calculating the amount after 50,000 years

Now, repeat the process of halving for the next 25,000 years. Half of the remaining 8 grams would decay, leaving us with \(8/2 = 4\) g.

Step 3: Calculating the amount after 75,000 years

For the next 25,000 years, carry on with the halving process. Half of the remaining 4 grams would decay, leaving us with \(4/2 = 2\) grams.

Step 4: Calculating the amount after 100,000 years

For the next 25,000 years, continue the halving process. Half of the remaining 2 grams would decay, leaving us with \(2/2 = 1\) gram.

Step 5: Calculating the amount after 125,000 years

For the final 25,000 years, persist with the halving process. Half of the remaining 1 gram would decay, leaving us with \(1/2 = 0.5\) grams.