The half-life of lead is 22 years. How long will it take for a sample of this substance to

decay to 80% of its original amount

A = Ao * (1/2)^(n/22)

.8 = .5^(n/22)

log .8 = (n/22) log .5

n/22 = .3219

n =7.08 years

Well, let me lead you to the punchline. To calculate this, we need to consider that after one half-life, the sample will decay to half of its original amount. So, after the first half-life, it will be at 50% of its original amount. Now, we want to know how long it will take for the sample to decay to 80% of its original amount. So, we need to find out how many half-lives it takes for it to go from 100% to 80%. If each half-life is 22 years, let me do some math here with my clown calculator... Ah, I got it! Since 50% is half of 100%, and only one half-life is needed for that, we need to divide the remaining 30% (80% - 50%) in half. That means, it will take another half-life to go from 50% to 75%, and yet another half-life to go from 75% to 87.5%. So, to reach 80%, we need approximately 2.5 half-lives. Therefore, it will take around 55 years (2.5 x 22 years) for the sample of lead to decay to 80% of its original amount.

To calculate the time it takes for a sample of lead to decay to 80% of its original amount, we can use the concept of half-life.

The half-life of lead is 22 years, which means that in 22 years, the amount of lead in a sample will decrease by half.

To find out how long it takes for the sample to decay to 80% of its original amount, we need to determine how many half-lives are required to reach 80%.

Let's do the calculation step-by-step:

1. Determine the number of half-lives required to reach 80%:
- 80% is equal to 0.8 (since percentages are written as decimals)
- To find the number of half-lives, we need to solve the equation (1/2)^(number of half-lives) = 0.8
- Taking the logarithm of both sides, we get: log((1/2)^(number of half-lives)) = log(0.8)
- Using the logarithmic property, we can bring down the exponent: number of half-lives * log(1/2) = log(0.8)
- Dividing both sides by log(1/2), we get: number of half-lives = log(0.8) / log(1/2)

2. Calculate the number of half-lives:
- Using a calculator, we can find: number of half-lives ≈ 0.3219 / -0.3010
- This gives us: number of half-lives ≈ 1.07

3. Calculate the time it takes to reach 80%:
- Since each half-life is 22 years, we can multiply the number of half-lives by 22 to get the time in years:
- Time = number of half-lives * 22 = 1.07 * 22 ≈ 23.54 years

Therefore, it will take approximately 23.54 years for a sample of lead to decay to 80% of its original amount.

To determine how long it will take for a sample of lead to decay to 80% of its original amount, we need to calculate the number of half-lives it will undergo.

The formula to calculate the number of half-lives is:

Number of half-lives = (log(final amount) - log(initial amount)) / log(0.5)

In this case, the final amount is 80% (or 0.8) of the original amount, and the initial amount is 100% (or 1) of the original amount.

Using the formula above, we can calculate the number of half-lives:

Number of half-lives = (log(0.8) - log(1)) / log(0.5)

Now, let's calculate this using a scientific calculator or a programming language:

log(0.8) = -0.09691001300805642 (approximately)
log(1) = 0 (log of 1 is always 0)
log(0.5) = -0.3010299956639812 (approximately)

Number of half-lives = (-0.09691001300805642 - 0) / -0.3010299956639812

Number of half-lives ≈ 0.322 (approximately)

Since we cannot have a fraction of a half-life, we can conclude that it will take one half-life for the sample of lead to decay to 80% of its original amount.

To find the time it takes for one half-life, we multiply the half-life of lead by the number of half-lives:

Time for one half-life ≈ 22 years × 0.322

Time for one half-life ≈ 7.0844 years (approximately)

Therefore, it will take approximately 7.0844 years for a sample of lead with a half-life of 22 years to decay to 80% of its original amount.