How do I write an equation for this problem?

[disparate]

The half-life of uranium is 1000 years. If 50 grams of uranium is sealed in a box, how much is left after 10,000 years? How long will it take to reduce to 1% of the original amount?


I know it's going to include logarithms, but I'm not exactly sure how to set up an equation for it. Is it 10,000=1000(50^x)?

 

[HallsofIvy]

If you start with A grams, in 1000 years you will have half of it left: (1/2)A. In another 1000 years, 1/2 of that, (1/2)((1/2)A)= (1/2)^2 A grams. In other words, since the half life is 1000 years, every 1000 years, multiply by 1/2. In "t" years, there are t/1000 periods of 1000 years- so A will be multiplied by 1/2 t/1000 times: A times \(\displaystyle (1/2)^{t/1000}\).

 

[disparate]

If you start with A grams, in 1000 years you will have half of it left: (1/2)A. In another 1000 years, 1/2 of that, (1/2)((1/2)A)= (1/2)^2 A grams. In other words, since the half life is 1000 years, every 1000 years, multiply by 1/2. In "t" years, there are t/1000 periods of 1000 years- so A will be multiplied by 1/2 t/1000 times: A times \(\displaystyle (1/2)^{t/1000}\).

So if I'm trying to solve for the amount of uranium after 10,000 years, would my equation be 50 x 1/2^10,000/1000?

Never mind, I've solved it.

 

[HallsofIvy]

Great. I presume that you have also solved the second part- there will be 1% of the original amount left when \(\displaystyle 50\left(\frac{1}{2}\right)^{t/1000}= .01(50)\) or \(\displaystyle \left(\frac{1}{2}\right)^{t/1000}= .01\). Solve that by taking the logarithm of both sides.