May someone help me with my HW problem (b)?
Radiocarbon dating can be used to estimate the age of material, such as
animal bones or plant remains, that comes from a formerly living organism.
When the organism dies, the amount of carbon-14 in its remains decays
exponentially. Suppose that f(t) is the amount of carbon-14 in an organism
at time t (in years), where t = 0 corresponds to the time of the death of the
organism. Assume that the amount of carbon-14 is modelled by the
exponential decay function
f(t) = Ce^kt (t >= 0);
where C is the initial amount of carbon-14 and k is a constant.
The half-life (or halving period) of a radioactive substance like carbon-14 is
the time that it takes for the amount of the substance to decrease to half of
its original level.
(a) Given that the half-life of carbon-14 is 5730 years, show that the value
of the constant k is -0.000121, correct to three significant figures.
(b) Suppose that some ancient animal bones have been found where the
amount of carbon-14 is known to have decreased to 1/5 of the level that
was present immediately after the death of the animal. How much time
has elapsed since the death of this animal? Give your answer to the
nearest whole number of years.
(a) Since we're dealing with half-life, the initial equation is
f(t) = C(1/2)^(t/5730)
Now, 1/2 = e^(ln 1/2) = e^(-ln2)
So,
f(t) = C*(e^-ln2)^(t/5730) = C*e^(-ln2/5730 t) = Ce^(-0.000121t)
(b) 1/5 = e^(-0.000121t)
ln(1/5) = -0.000121t
Now finish it off
Thanks for your help
To solve part (b) of your problem, we need to use the exponential decay function for carbon-14:
f(t) = Ce^(kt)
We know that for the given problem, the amount of carbon-14 has decreased to 1/5 of its initial level. So, we can say that:
f(t) = (1/5)C
We want to find the time, t, that has elapsed since the death of the animal. To do this, we need to isolate t in the equation above.
Step 1: Plug in the given value of k.
We were previously asked to find the value of the constant k in part (a). This value was found to be -0.000121, correct to three significant figures. So, we substitute this value into our equation:
(1/5)C = Ce^(-0.000121t)
Step 2: Cancel out the C terms.
We can divide both sides of the equation by C to eliminate it from the equation:
(1/5) = e^(-0.000121t)
Step 3: Take the natural logarithm of both sides.
We want to isolate t, so we remove the exponential function by taking the natural logarithm (ln) of both sides of the equation:
ln(1/5) = ln(e^(-0.000121t))
Step 4: Use logarithm properties.
By using the property that ln(e^x) = x, the equation simplifies to:
ln(1/5) = -0.000121t
Step 5: Solve for t.
Finally, divide both sides of the equation by -0.000121 to solve for t:
t = ln(1/5) / -0.000121
Now, you can use a calculator to find the value of t. Evaluating this expression gives us t ≈ 4153.09.
Since we want the answer to the nearest whole number of years, the time that has elapsed since the death of the animal is approximately 4153 years.