Final answer:
To determine how long 12,000 lbs of a substance with a half-life of 15,000 years will take to decay to 10,000 lbs, we use the concept of exponential decay and set up the equation N(t) = N0(1/2)ᵗ/ᵀ, solve for t, and find the time required.
Step-by-step explanation:
The half-life of a substance refers to the time required for half of the material to undergo decay radioactively. To find out how long it will take for 12,000 lbs of a chemical compound to decay to 10,000 lbs, we can apply the concept of half-life.
First, we need to determine the fraction of the original substance that 10,000 lbs represents. In this case, it's 10,000 lbs / 12,000 lbs = 5/6.
Since the half-life is the time it takes for half of the substance to decay, we can use the formula for exponential decay:
N(t) = N0(1/2)ᵗ/ᵀ
where:
- N(t) is the remaining amount of substance after time t,
- N0 is the initial amount of substance,
- t is the time elapsed,
- T is the half-life period.
So, we set up the equation 10,000 = 12,000(1/2)⁽ᵗ/¹⁵,⁰⁰⁰⁾. If we divide both sides by 12,000, we get:
5/6 = (1/2)⁽ᵗ/¹⁵,⁰⁰⁰⁾
To solve for t, we take the natural logarithm of both sides and then solve for t:
ln(5/6) = (t/15,000) ln(1/2)
t = (ln(5/6) / ln(1/2)) * 15,000 years
After calculating this expression, we find the value of t, which represents how long it will take for the substance to go from 12,000 lbs to 10,000 lbs.