The value of the time is .
The correct option is
Further explanation:
Carbon-14 is a weakly radioactive isotope of carbon.
Radiocarbon dating is the archaeological process to determine the age of an object which should be of organic material with the properties of radiocarbon.
The radiocarbon decay is the amount of carbon which is converted into wood.
The amount of the substance at any time is calculated as follows:
......(1)
Here, is the initial amount of the substance and is the constant.
The concentration in the initial amount is always .
Given:
The half-life of the Carbon-14 is . The concentration of the carbon in the piece of wood is .
Calculation:
Step 1:
First we calculate the rate constant of the reaction.
The half-life of Carbon-14 is .
Substitute for and for in the equation (1) to obtain the value of .
Further solve the above equation to obtain the value of .
Therefore, the value of is .
Step 2:
Now we calculate the age of wood.
The concentration of initial carbon is .
The amount of the carbon remaining in the wood is which is the final concentration.
The value of is obtained from the equation (1) as follows:
Therefore, the expression for is .
Now, substitute for , for and for in the expression of .
Therefore, the value of is .
Thus, the age of sample of Carbon-14 is .
This implies that the correct option is .
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Answer details:
Grade: Junior school
Subject: Mathematics
Chapter: Exponential function
Keywords: Half-life, carbon-14, isotope, reaction, wood, concentration, amount, initial amount, final point, time, years, age of wood, exponents, scientific form, radiocarbon. Radiocarbon decay, carbon dating.