Problem 45
Question
Carbon- 14 is a radioactive isotope used to date objects. If \(A_{0}\) represents the initial amount of carbon- 14 in the object, then the quantity remaining at time \(t\), in years, is $$ A(t)=A_{0} e^{-0.000121 t} $$ (a) A tree, originally containing 185 micrograms of carbon- 14 , is now 500 years old. At what rate is the carbon- 14 decaying? (b) In 1988 , scientists found that the Shroud of Turin, which was reputed to be the burial cloth of Jesus, contained \(91 \%\) of the amount of carbon- 14 in freshly made cloth of the same material. \({ }^{6}\) According to this data, how old was the Shroud of Turin in 1988 ?
Step-by-Step Solution
Verified Answer
(a) -0.0112 micrograms/year. (b) The Shroud of Turin is approximately 813 years old, originating around 1175.
1Step 1 - Calculate Carbon-14 Decay Rate for the Tree
The decay rate of the carbon-14 is given by the derivative of the function \( A(t) = A_{0} e^{-0.000121 t} \). Differentiate \( A(t) \) with respect to \( t \):\[ \frac{dA}{dt} = -0.000121 A_{0} e^{-0.000121 t}. \] Now substitute \( A_{0} = 185 \) micrograms and \( t = 500 \) years into the differentiated equation: \[ \frac{dA}{dt} = -0.000121 \times 185 \times e^{-0.000121 \times 500}. \] This will give the decay rate at \( t = 500 \) years.
2Step 2 - Substitute and Solve for the Rate
Substitute the values into the derivative equation: \[ \frac{dA}{dt} = -0.000121 \times 185 \times e^{-0.0605}. \] Calculate \( e^{-0.0605} \) and use it to find: \[ \frac{dA}{dt} \approx -0.0112 \text{ micrograms/year.} \] The negative sign indicates decay.
3Step 3 - Determine Age of the Shroud of Turin
We know \( A(t) = 0.91 A_{0} \) which implies \( 0.91 = e^{-0.000121 t} \). Take the natural logarithm on both sides to solve for \( t \): \[ \ln(0.91) = -0.000121 t. \] Solve for \( t \) to determine the age of the shroud: \[ t = \frac{\ln(0.91)}{-0.000121}. \] Calculate to find \( t \approx 813 \) years.
4Step 4 - Calculate Year of Origin for the Shroud
If the shroud was 813 years old in 1988, subtract 813 from 1988 to find the estimated year of origin.\[ 1988 - 813 = 1175. \] Thus, the shroud is estimated to have originated around the year 1175.
Key Concepts
Radioactive DecayExponential FunctionsDerivative Calculation
Radioactive Decay
Radioactive decay is a process in which an unstable radioactive isotope transforms into a more stable element over time. This occurs at a fixed rate, specific to each radioactive material. In our exercise, we focus on Carbon-14, an isotope widely used for dating archaeological and geological samples.
Carbon-14 is produced in the upper atmosphere and absorbed by living organisms. After the organism dies, Carbon-14 starts to decay, which lets us measure how long it has been since the organism passed away.
The decay rate for any radioactive material, like Carbon-14, can be modeled using exponential decay functions. It decreases over time exponentially, which is why it appears in our formula for this exercise. The key number here is the decay constant. For Carbon-14, it's -0.000121, which tells us the percentage decay per year. This constant gives us the ability to estimate the age of objects containing Carbon-14.
Carbon-14 is produced in the upper atmosphere and absorbed by living organisms. After the organism dies, Carbon-14 starts to decay, which lets us measure how long it has been since the organism passed away.
The decay rate for any radioactive material, like Carbon-14, can be modeled using exponential decay functions. It decreases over time exponentially, which is why it appears in our formula for this exercise. The key number here is the decay constant. For Carbon-14, it's -0.000121, which tells us the percentage decay per year. This constant gives us the ability to estimate the age of objects containing Carbon-14.
Exponential Functions
Exponential functions are mathematical expressions where a constant base is raised to the power of a variable. In the context of radioactive decay, this function describes how the amount of a substance decreases over time.
Our formula used here, \(A(t) = A_0 e^{-0.000121 t}\), is an exponential decay function. The base here is \(e\), an important irrational number approximately equal to 2.718. This function models the decay process as a continuously compounding phenomenon, capturing how the quantity decreases as years pass.
The initial amount \(A_0\) is the starting quantity of Carbon-14, and as time \(t\) increases, the exponential term \(e^{-0.000121 t}\) decreases, resulting in diminishing values of \(A(t)\). This model allows us to predict the remaining quantity of Carbon-14 at any given time and, importantly, to estimate the age of objects based on the remaining C-14 content.
Our formula used here, \(A(t) = A_0 e^{-0.000121 t}\), is an exponential decay function. The base here is \(e\), an important irrational number approximately equal to 2.718. This function models the decay process as a continuously compounding phenomenon, capturing how the quantity decreases as years pass.
The initial amount \(A_0\) is the starting quantity of Carbon-14, and as time \(t\) increases, the exponential term \(e^{-0.000121 t}\) decreases, resulting in diminishing values of \(A(t)\). This model allows us to predict the remaining quantity of Carbon-14 at any given time and, importantly, to estimate the age of objects based on the remaining C-14 content.
Derivative Calculation
Calculating a derivative involves finding the rate at which a function changes. It's like figuring out how fast a car is going at any given moment, based on its changing position over time.
In this exercise, we calculate the derivative of the decay function \(A(t) = A_0 e^{-0.000121 t}\) to determine how fast the Carbon-14 is decaying. The derivative formula \(\frac{dA}{dt} = -0.000121 A_0 e^{-0.000121 t}\) tells us the rate of this change. Here, \(-0.000121\) is retained as a coefficient since it's our decay constant.
The decay rate can indicate how quickly the material is losing its Carbon-14 content. When substituting the initial amount and specific time, such as 500 years for a tree in the problem, we find the specific decay rate at that moment, helping us understand the current status of the decay process.
Applying this concept, we can calculate rates of change in various scientific inquiries, making derivatives a powerful mathematical tool beyond just radioactive decay.
In this exercise, we calculate the derivative of the decay function \(A(t) = A_0 e^{-0.000121 t}\) to determine how fast the Carbon-14 is decaying. The derivative formula \(\frac{dA}{dt} = -0.000121 A_0 e^{-0.000121 t}\) tells us the rate of this change. Here, \(-0.000121\) is retained as a coefficient since it's our decay constant.
The decay rate can indicate how quickly the material is losing its Carbon-14 content. When substituting the initial amount and specific time, such as 500 years for a tree in the problem, we find the specific decay rate at that moment, helping us understand the current status of the decay process.
Applying this concept, we can calculate rates of change in various scientific inquiries, making derivatives a powerful mathematical tool beyond just radioactive decay.
Other exercises in this chapter
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