Carbon-14 has a half-life of about 5,600 years. Archaeologists analyze a piece of wood found at an ancient village site and determine that, of he initial 100 grams of Carbon-14 in the wood, onl 3.125 grams are left. How old is the wood?

3.125=100(1/2)^n/thalf

take log both sides...

log(3.125)=2+n/thlaf (log.5)
so solve for n, then grab you calculator and have fun.

8898

To determine the age of the wood, we can use the formula for exponential decay:

N(t) = N0 * (1/2)^(t / t(1/2))

Where:
N(t) = the amount of substance remaining after time t
N0 = the initial amount of substance
t(1/2) = the half-life of the substance

In this case, we are given that the initial amount of Carbon-14 (N0) is 100 grams and the remaining amount (N(t)) is 3.125 grams.

3.125 = 100 * (1/2)^(t / 5600)

Now, we can solve for t, the age of the wood. Let's isolate the exponential term:

(1/2)^(t / 5600) = 3.125 / 100

Simplifying the right side:

(1/2)^(t / 5600) = 0.03125

Next, let's take the logarithm of both sides to simplify the equation further:

log((1/2)^(t / 5600)) = log(0.03125)

Using the property of logarithms (log(a^b) = b * log(a)):

(t / 5600) * log(1/2) = log(0.03125)

Now, let's solve for t by isolating it:

t / 5600 = log(0.03125) / log(1/2)

Using the logarithmic properties again:

t = (5600 * log(0.03125)) / log(1/2)

Using a calculator, we can evaluate the logarithms:

t ≈ (5600 * (-1.50514997832)) / (-0.30102999566) ≈ 28,799.98

Therefore, the wood is approximately 28,799.98 years old.

To determine the age of the wood, we can use the concept of carbon dating, which relies on the decay of carbon-14 in organic materials. Carbon-14 has a known half-life of about 5,600 years, which means that after 5,600 years, half of the carbon-14 in a sample will have decayed.

In this case, we are given that there is 3.125 grams of carbon-14 remaining out of the initial 100 grams. To find the age of the wood, we need to determine how many half-lives have passed.

We can set up an equation to solve for the number of half-lives:

3.125 grams = 100 grams × (1/2)^(number of half-lives)

To isolate the number of half-lives, we can take the logarithm (base 2) of both sides of the equation:

log2(3.125) = log2(100) + (number of half-lives) × log2(1/2)

Using a calculator, we can find that log2(3.125) is approximately 1.678. Since log2(1/2) is -1, and log2(100) is approximately 6.643, we have:

1.678 = 6.643 + (number of half-lives) × (-1)

Simplifying the equation:

1.678 = 6.643 - number of half-lives

Rearranging the equation to isolate the number of half-lives:

number of half-lives = 6.643 - 1.678

number of half-lives = 4.965

Since each half-life corresponds to approximately 5,600 years, we can multiply the number of half-lives by the length of each half-life to find the age of the wood:

age of the wood ≈ 4.965 × 5,600 years

Therefore, the wood is approximately 27,864 years old.