Answered

Discover answers to your questions with Westonci.ca, the leading Q&A platform that connects you with knowledgeable experts. Get expert answers to your questions quickly and accurately from our dedicated community of professionals. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.

Suppose that the ratio of carbon-14 to carbon-12 in a piece of wood discovered in a cave is [tex] R = \frac{1}{8^{16}} [/tex]. Write an equation in terms of [tex] t [/tex] that can be used to determine the age of the piece of wood.

[tex]\[ (-8223) \frac{\ln \left(2^{-48}\right)}{\ln (e)}=t \][/tex]

(Note: The equation given seems incorrect or unclear. The decay constant for carbon-14 should be used in this context. Let's rewrite it for clarity and accuracy.)

Corrected Response:

Suppose that the ratio of carbon-14 to carbon-12 in a piece of wood discovered in a cave is [tex] R = \frac{1}{8^{16}} [/tex]. Write an equation in terms of [tex] t [/tex] that can be used to determine the age of the piece of wood.

Given:
[tex]\[ R = \frac{1}{8^{16}} \][/tex]

We know that the decay of carbon-14 is given by:
[tex]\[ R = e^{-\lambda t} \][/tex]

where [tex] \lambda [/tex] is the decay constant for carbon-14.

Rewriting the equation in terms of [tex] t [/tex]:
[tex]\[ t = \frac{\ln(R)}{-\lambda} \][/tex]

Substitute [tex] R = \frac{1}{8^{16}} \]:
[tex]\[ t = \frac{\ln\left(\frac{1}{8^{16}}\right)}{-\lambda} \][/tex]

Given the relationship [tex] 8^{16} = 2^{48} \], we have:
[tex]\[ \ln\left(\frac{1}{8^{16}}\right) = \ln\left(2^{-48}\right) = -48 \ln(2) \][/tex]

Thus,
[tex]\[ t = \frac{-48 \ln(2)}{-\lambda} = \frac{48 \ln(2)}{\lambda} \][/tex]

So, the equation in terms of [tex] t [/tex] is:
[tex]\[ t = \frac{48 \ln(2)}{\lambda} \][/tex]

where [tex] \lambda [/tex] is the decay constant for carbon-14.

Sagot :

GinnyAnswer
Let's break down and solve this problem step by step.

### Step 1: Understand the given information

Importantly, we know:
- The decay formula for carbon-14 is given by the relationship \( R = \left( \frac{1}{2} \right)^{\frac{t}{5730}} \), where \( R \) is the ratio of carbon-14 to carbon-12, \( t \) is the time in years, and 5730 years is the half-life of carbon-14.
- The ratio \( R = \frac{1}{8^{16}} \).

### Step 2: Setting up the equation
Given \( R = \left( \frac{1}{2} \right)^{\frac{t}{5730}} \) and \( R = \frac{1}{8^{16}} \), we equate the two expressions for \( R \):

[tex]\[ \left( \frac{1}{2} \right)^{\frac{t}{5730}} = \frac{1}{8^{16}} \][/tex]

### Step 3: Transform the right-hand side expression for simplicity
We rewrite \(\frac{1}{8^{16}}\) to match the left-hand side form:

[tex]\[ 8 = 2^3 \implies 8^{16} = (2^3)^{16} = 2^{48} \implies \frac{1}{8^{16}} = \frac{1}{2^{48}} = 2^{-48} \][/tex]

So the equation becomes:

[tex]\[ \left( \frac{1}{2} \right)^{\frac{t}{5730}} = 2^{-48} \][/tex]

### Step 4: Solve for \( t \) using logarithms
We take the natural logarithm (\(\ln\)) of both sides to solve for \( t \):

[tex]\[ \ln \left( \left( \frac{1}{2} \right)^{\frac{t}{5730}} \right) = \ln (2^{-48}) \][/tex]

Using the logarithm property \(\ln (a^b) = b \ln (a)\), we get:

[tex]\[ \frac{t}{5730} \ln \left( \frac{1}{2} \right) = \ln (2^{-48}) \][/tex]

Since \(\ln \left( \frac{1}{2} \right) = -\ln (2)\) and \(\ln (2^{-48}) = -48 \ln (2)\), the equation simplifies to:

[tex]\[ \frac{t}{5730} (-\ln (2)) = -48 \ln (2) \][/tex]

### Step 5: Isolate \( t \)
We can now solve for \( t \):

[tex]\[ t = \frac{-48 \ln (2) \cdot 5730}{-\ln (2)} \][/tex]

The \(\ln (2)\) cancels out:

[tex]\[ t = 48 \cdot 5730 \][/tex]

### Conclusion
Thus, the age \( t \) of the piece of wood is:

[tex]\[ t = 275040 \][/tex]

Rewriting this in the form given in the problem:

[tex]\[ t = (-8223) \frac{\ln \left(2^{-48}\right)}{\ln (e)} = 275040 \text{ years} \][/tex]

This completes our detailed step-by-step solution for determining the age of the piece of wood given the provided ratio of carbon-14 to carbon-12.
Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.