A gram of living plant material with Carbon-14 , decays at a rate of about 16 disintegrations per minute. When the plant dies, the Carbon-14 decays, with a half-life of 5600 years. A gram of the dead plant is found to decay at about 1 disintegration per minute. Calculate the probable age of the plant.
rate = initial rate e^-kt
1/2 = e^-k(5600)
ln .5 = -5600 k
-.693147 = - 5600 k
so
k = 1.2378 *10^-4
now
1 = 16 e^-1.2378 *10^-4 t
ln 1/16 = -1.2378 *10^-4 t
t = 22,400
or do it in your head
1/16 = (1/2)^4
so 4 * 5600 = 22400 :) LOL
Well, well, well! It seems like we've stumbled upon a carbon dating conundrum! Don't worry, I'm here to help you out with some math (and laughter).
Let's break it down step by step. We know that the living plant material decays at 16 disintegrations per minute, while the dead plant decays at 1 disintegration per minute.
The half-life of Carbon-14 is 5600 years. This means that it takes 5600 years for half of the Carbon-14 in a sample to decay.
Since the living plant material decays at 16 disintegrations per minute, we can assume that it contains a lot more Carbon-14 than the dead plant material, which only decays at 1 disintegration per minute.
To find the age of the plant, we need to calculate how many half-lives have passed between the time the plant was alive and when it was found dead.
So, let's do some math (don't worry, I'll keep it funny):
- If the living plant material decays at 16 disintegrations per minute, that means it's approximately 16 times more "alive" than the dead plant, right? Talk about being a lively bunch!
- So, if the dead plant decays at 1 disintegration per minute, we can say that it's relatively "dead" compared to the living plant material. Poor thing!
Now, let's compare the two rates and see how many half-lives have passed:
16 disintegrations per minute (living) -> 1 disintegration per minute (dead)
By dividing the decay rate of the living plant material by the decay rate of the dead plant material, we can determine the number of half-lives that have passed:
16 / 1 = 16 half-lives
Each half-life is 5600 years, so:
16 half-lives * 5600 years per half-life = 89,600 years
So, based on our calculations, the probable age of the dead plant material is approximately 89,600 years old. Isn't it amazing how nature can make us laugh while also revealing its secrets?
To calculate the probable age of the plant, we can use the concept of half-life. The half-life of Carbon-14 is 5600 years, meaning that after every 5600 years, the amount of Carbon-14 in a sample is halved.
Let's break down the information given:
1. When the plant is alive, it decays at a rate of 16 disintegrations per minute.
2. When the plant dies, it decays at a rate of 1 disintegration per minute.
We can use the ratio between the decay rates to determine the age of the plant.
The ratio of the decay rates is:
(16 disintegrations per minute) / (1 disintegration per minute)
This ratio tells us how many times faster the Carbon-14 is decaying when the plant is alive compared to when it is dead.
Simplifying this ratio, we get:
16 / 1 = 16
Now, we need to determine how many half-lives have passed based on this ratio.
To do this, we take the logarithm (base 2) of the ratio:
log2(16) = 4
The result, 4, represents the number of half-lives that have passed since the plant died.
Since each half-life of Carbon-14 is 5600 years, we can calculate the probable age of the plant:
4 half-lives * 5600 years per half-life = 22400 years
Therefore, the probable age of the plant is 22400 years.
To calculate the probable age of the plant, we can use the concept of half-life and the given decay rates.
First, let's calculate the decay constant (λ) for Carbon-14 using its half-life:
λ = ln(2) / t₁/₂
where ln refers to the natural logarithm and t₁/₂ is the half-life.
In this case, t₁/₂ = 5600 years, so we have:
λ = ln(2) / 5600
Using this decay constant, we can calculate the initial amount of Carbon-14 in a gram of living plant material.
Let N₀ be the initial amount of Carbon-14, and N(t) be the amount remaining at time t, then the decay equation can be expressed as:
N(t) = N₀ * e^(-λt)
where e is Euler's number (approximately 2.718).
Given that the decay rate of living plant material is 16 disintegrations per minute, we can substitute this information to obtain:
16 = N₀ * e^(-λ * t),
where t is the time in minutes.
Now, for the dead plant material, the decay rate is 1 disintegration per minute. We can use the same equation, but with the new decay rate, to find the time t for the dead plant material:
1 = N₀ * e^(-λ * t).
Now we can solve the two equations to find the time t, which will give us the age of the plant.
To do this, we divide the two equations:
(16 / 1) = (N₀ * e^(-λ * t)) / (N₀ * e^(-λ * t)),
giving us:
16 = e^(-λ * t) / e^(-λ * t),
which simplifies to:
16 = 1.
Since this equation is not possible, we conclude that there is an error in the given information. The decay rates of the living and dead plant materials should follow the same exponential decay curve, indicating that the age of the plant cannot be accurately determined with the provided information.