# Solve the following exponential application problem involving half life of a substance. Plutonium-239 has a half-life of 24,000 years. A rule of thumb is that radioactive wastes are virtually...

Solve the following exponential application problem involving half life of a substance.

Plutonium-239 has a half-life of 24,000 years. A rule of thumb is that radioactive wastes are virtually harmless after 10 half-lives. How long must 1 gram of Plutonium-239 be securely stored before it is virtually harmless.

you have to use the equation A= A02-t/k to solve. Also the answer to this problem is 240,000, but I am not looking for the answer, I need help with how to get to the answer using the given equation. Please help.

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### 2 Answers

You haven't been able to understand the response I gave you earlier. Let me try to clarify. Plutonium 239 has a half life of 24000 years. Half life is the duration of time required for the initial amount of a substance to reduce to 1/2 the amount. Plutonium is virtually harmless after 10 half lives. After 10 half lives, the initial amount reduces to 1/(2^10).

Now if you want to use the equation you have, you have to plug in the value A0*(1/2)^10 as the final amount.

This gives A0(1/2)^10 = A0*2^(-t/ 24000)

=> (1/2)^10 = 2^(-t/24000)

=> 2^10 = 2^(t/24000)

=> 10 = t/24000

=> t = 10*24000

=> t = 240000

But here you are actually using the answer to create the value that has to be used in the equation and then using the equation to get back the same value that started with initially.

Instead of doing all this, you should learn when to use the equation and when that is not required. Here, the use of the equation is not required as you can write the answer straight away from the information provided.

We'll use the half life formula to determine the rate decay.

24000 = -ln 2/k

k = -ln 2/24000

k = -0.000029

Now, we'll determine the rate of decay for 1 g of Plutonium.

1 = 10e^-0.000029*t

ln 0.1 = ln e^-0.000029t

ln 0.1 = -0.000029t

t = -0.000029/ln 0.1

**The time needed by the 1 g of Plutonium to be securelly stored is approx. 80 000 years.**