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Answer:
<em>Carbon-14 will take 19,035 years to decay to 10 percent.</em>
Step-by-step explanation:
<u>Exponential Decay Function</u>
A radioactive half-life refers to the amount of time it takes for half of the original isotope to decay.
An exponential decay can be described by the following formula:
Where:
No = The quantity of the substance that will decay.
N(t) = The quantity that still remains and has not yet decayed after a time t
= The decay constant.
One important parameter related to radioactive decay is the half-life:
If we know the value of the half-life, we can calculate the decay constant:
Carbon-14 has a half-life of 5,730 years, thus:
The equation of the remaining quantity of Carbon-14 is:
We need to calculate the time required for the original amout to reach 10%, thus N(t)=0.10No
Simplifying:
Taking logarithms:
Solving for t:
Carbon-14 will take 19,035 years to decay to 10 percent.
For this case we must solve the following quadratic equation:
Where:
The solution is given by:
Substituting:
Thus, we have two roots:
ANswer: