Please help me solve this question:

A layer of peat beneath the glacial sediments of the last ice age had a carbon-14 content of 25% of that found in living organisms. How long ago was this ice age?

You will need k and you don't have information to get it. Look up the half life of C-14 which I think is in the neighborhood of 5800 years. Use the exact number when you find it.

Then k = 0.693/t1/2

ln(No/N) = kt
Substitute 100 for No.
N = 25
k from the half life
Solve for t.

To determine how long ago the ice age occurred, we need to understand the concept of carbon-14 dating and its decay rate.

Carbon-14 is a radioactive isotope that is naturally present in living organisms, and its half-life is approximately 5730 years. This means that after 5730 years, only half of the original amount of carbon-14 remains.

In the question, it is mentioned that the peat layer had a carbon-14 content of 25% of that found in living organisms. This implies that the remaining 75% has decayed since the time of the ice age.

Let's denote the original amount of carbon-14 in the peat layer as 100%. After the decay, we are left with 25% of the original amount. Using this information, we can set up an equation:

(100%) * (0.5)^n = (25%)

In this equation, 'n' represents the number of half-lives that have passed since the ice age.

To solve for 'n,' we can take the logarithm of both sides of the equation to eliminate the exponent:

log(0.5^n) = log(0.25)
n * log(0.5) = log(0.25)

Using the properties of logarithms, we can simplify the equation further:

n = log(0.25) / log(0.5)

Evaluating this expression using a calculator, we get:

n ≈ 2.3219

Since 'n' represents the number of half-lives, we can round this value to 2, as we cannot have a fraction of a half-life. Therefore, approximately 2 half-lives have passed since the ice age.

To find the time elapsed, we multiply the number of half-lives by the half-life of carbon-14:

Time elapsed ≈ 2 * 5730 years

Calculating this, we find:

Time elapsed ≈ 11,460 years

Therefore, the ice age occurred approximately 11,460 years ago.

To solve this question, we need to understand the concept of carbon-14 dating. Carbon-14 (or radiocarbon) dating is a method used to determine the age of organic materials based on the decay rate of carbon-14 isotopes.

The first step is to recognize that living organisms have a normal amount of carbon-14 in their bodies. When an organism dies, it no longer takes in any new carbon-14, and over time, the carbon-14 in its remains undergoes radioactive decay with a known half-life of about 5730 years.

Now, let's approach the question.

Given that the layer of peat beneath the glacial sediments had a carbon-14 content of 25% of that found in living organisms, it means that after the ice age, there was still some carbon-14 left in the peat, albeit reduced to 25% compared to a living organism.

To determine how long ago the ice age happened, we can use the half-life of carbon-14 to calculate the number of half-lives that occurred to reach 25% of its original content.

Since the radioactive decay of carbon-14 occurs with a half-life of 5730 years, one half-life would reduce the carbon-14 content by half, and two half-lives would reduce it to a quarter (25%).

Let's calculate the number of half-lives that occurred:

Number of Half-lives = Logarithm(Base 0.5)(Final amount/Initial amount)

Number of Half-lives = Logarithm(0.5)(0.25/1)

Number of Half-lives ≈ 2

Therefore, two half-lives have occurred for the carbon-14 content in the peat layer to reach 25% of its original amount.

To determine the time it took for these two half-lives, we multiply the half-life duration by the number of half-lives:

Time = Half-life x Number of Half-lives

Time = 5730 years x 2

Time ≈ 11,460 years

Therefore, the ice age occurred approximately 11,460 years ago.

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