Question
Carbon-14 has a half-life of 5730 years. A sample of carbon-14 was found during an archaeological dig which had a weight of 5 mg. It is known that the original size of the sample was 20 mg.
Which statement about this situation is TRUE?
The equation 20= 5(1/2)^5730x
models this situation. The sample is 0.00035 years old.
The equation 20= 5(1/2)^x/5730
models this situation. The sample is 11460 years old.
The equation 5= 20(1/2)^x/5730
models this situation. The sample is 11460 years old.
The equation 5=20(1/2)5730x
models this situation. The sample is 0.00035 years old.
Which statement about this situation is TRUE?
The equation 20= 5(1/2)^5730x
models this situation. The sample is 0.00035 years old.
The equation 20= 5(1/2)^x/5730
models this situation. The sample is 11460 years old.
The equation 5= 20(1/2)^x/5730
models this situation. The sample is 11460 years old.
The equation 5=20(1/2)5730x
models this situation. The sample is 0.00035 years old.
Answers
GPT-4o mini
To analyze the statements provided, we need to understand the relationship between the original amount of Carbon-14, the remaining amount, and the half-life.
Given:
- Half-life of Carbon-14 = 5730 years
- Original amount of the sample = 20 mg
- Remaining amount of the sample = 5 mg
The formula to determine the amount of substance remaining after a certain time can be modeled by the equation:
\[
N = N_0 \times \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}}
\]
Where:
- \(N\) is the remaining amount of substance (5 mg)
- \(N_0\) is the initial amount of substance (20 mg)
- \(t\) is the time elapsed
- \(T_{1/2}\) is the half-life (5730 years)
Plugging in the known values, we rearrange the formula to find \(t\):
\[
5 = 20 \times \left( \frac{1}{2} \right)^{\frac{t}{5730}}
\]
Dividing both sides by 20 gives:
\[
\frac{5}{20} = \left( \frac{1}{2} \right)^{\frac{t}{5730}}
\]
which simplifies to:
\[
\frac{1}{4} = \left( \frac{1}{2} \right)^{\frac{t}{5730}}
\]
Noting that \(\frac{1}{4} = \left( \frac{1}{2} \right)^{2}\), we can set the exponent equal to 2:
\[
\frac{t}{5730} = 2
\]
This implies:
\[
t = 2 \times 5730 = 11460 \text{ years}
\]
Now let's consult the provided options to find the correct statements:
1. **The equation 20= 5(1/2)^5730x models this situation. The sample is 0.00035 years old.**
- Incorrect (wrong equation and age).
2. **The equation 20= 5(1/2)^x/5730 models this situation. The sample is 11460 years old.**
- Correct (the equation can be rearranged to fit what we derived).
3. **The equation 5= 20(1/2)^x/5730 models this situation. The sample is 11460 years old.**
- Incorrect (the structure of the equation is not right).
4. **The equation 5=20(1/2)5730x models this situation. The sample is 0.00035 years old.**
- Incorrect (wrong equation and age).
Thus, the **true statement** is: **The equation 20= 5(1/2)^x/5730 models this situation. The sample is 11460 years old.**
Given:
- Half-life of Carbon-14 = 5730 years
- Original amount of the sample = 20 mg
- Remaining amount of the sample = 5 mg
The formula to determine the amount of substance remaining after a certain time can be modeled by the equation:
\[
N = N_0 \times \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}}
\]
Where:
- \(N\) is the remaining amount of substance (5 mg)
- \(N_0\) is the initial amount of substance (20 mg)
- \(t\) is the time elapsed
- \(T_{1/2}\) is the half-life (5730 years)
Plugging in the known values, we rearrange the formula to find \(t\):
\[
5 = 20 \times \left( \frac{1}{2} \right)^{\frac{t}{5730}}
\]
Dividing both sides by 20 gives:
\[
\frac{5}{20} = \left( \frac{1}{2} \right)^{\frac{t}{5730}}
\]
which simplifies to:
\[
\frac{1}{4} = \left( \frac{1}{2} \right)^{\frac{t}{5730}}
\]
Noting that \(\frac{1}{4} = \left( \frac{1}{2} \right)^{2}\), we can set the exponent equal to 2:
\[
\frac{t}{5730} = 2
\]
This implies:
\[
t = 2 \times 5730 = 11460 \text{ years}
\]
Now let's consult the provided options to find the correct statements:
1. **The equation 20= 5(1/2)^5730x models this situation. The sample is 0.00035 years old.**
- Incorrect (wrong equation and age).
2. **The equation 20= 5(1/2)^x/5730 models this situation. The sample is 11460 years old.**
- Correct (the equation can be rearranged to fit what we derived).
3. **The equation 5= 20(1/2)^x/5730 models this situation. The sample is 11460 years old.**
- Incorrect (the structure of the equation is not right).
4. **The equation 5=20(1/2)5730x models this situation. The sample is 0.00035 years old.**
- Incorrect (wrong equation and age).
Thus, the **true statement** is: **The equation 20= 5(1/2)^x/5730 models this situation. The sample is 11460 years old.**