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Sagot :
To create a graph of the radioactive decay of carbon-14 in terms of half-lives, follow these steps:
### Step 1: Convert Years to Number of Half-Lives
The half-life of carbon-14 is 5,700 years. To convert the number of years passed to the number of half-lives, divide each year value by 5,700.
Here are the conversions:
- 0 years: [tex]\( \frac{0}{5700} = 0 \)[/tex] half-lives
- 5,700 years: [tex]\( \frac{5700}{5700} = 1 \)[/tex] half-life
- 11,400 years: [tex]\( \frac{11400}{5700} = 2 \)[/tex] half-lives
- 17,100 years: [tex]\( \frac{17100}{5700} = 3 \)[/tex] half-lives
- 22,800 years: [tex]\( \frac{22800}{5700} = 4 \)[/tex] half-lives
- 28,500 years: [tex]\( \frac{28500}{5700} = 5 \)[/tex] half-lives
- 34,200 years: [tex]\( \frac{34200}{5700} = 6 \)[/tex] half-lives
- 39,900 years: [tex]\( \frac{39900}{5700} = 7 \)[/tex] half-lives
- 45,600 years: [tex]\( \frac{45600}{5700} = 8 \)[/tex] half-lives
- 51,300 years: [tex]\( \frac{51300}{5700} = 9 \)[/tex] half-lives
### Step 2: Plot the Data
We will use the converted half-lives as our [tex]\( x \)[/tex]-axis and the given amounts of carbon-14 as our [tex]\( y \)[/tex]-axis. Ensure the [tex]\( x \)[/tex]-axis is labeled "Number of half-lives" and the [tex]\( y \)[/tex]-axis is labeled "Amount of isotope (g)."
### Step 3: Create the Graph
To create a graph, follow these points:
1. Plot the points on a coordinate system.
2. Label the [tex]\( x \)[/tex]-axis as "Number of half-lives".
3. Label the [tex]\( y \)[/tex]-axis as "Amount of isotope (g)".
4. Draw a smooth curve through the points to represent the decay.
Here is the plotted data:
- (0, 10000)
- (1, 5000)
- (2, 2500)
- (3, 1250)
- (4, 625)
- (5, 312)
- (6, 156)
- (7, 78)
- (8, 39)
- (9, 20)
### Step 4: Example Plot
Imagine the graph as depicted below, but you can plot it on graph paper or use graphing software:
[tex]\[ \begin{array}{ccccccccc} \text{Number of half-lives (x)} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \text{Amount of isotope (g) (y)} & 10000 & 5000 & 2500 & 1250 & 625 & 312 & 156 & 78 & 39 & 20 \\ \end{array} \][/tex]
### Example Plot Description:
- The x-axis will show the number of half-lives ranging from 0 to 9.
- The y-axis will show the amount of isotope in grams, ranging from 0 to 10000.
- Each point will be plotted according to the match of half-lives and carbon-14 remaining.
- Connect the points to see the exponential decay pattern of the carbon-14 over time.
Once plotted, you will notice the exponential decay curve, indicating that the amount of carbon-14 decreases by half every 5,700 years.
This visual representation helps in understanding how radioactive decay progresses over multiple half-lives.
### Step 1: Convert Years to Number of Half-Lives
The half-life of carbon-14 is 5,700 years. To convert the number of years passed to the number of half-lives, divide each year value by 5,700.
Here are the conversions:
- 0 years: [tex]\( \frac{0}{5700} = 0 \)[/tex] half-lives
- 5,700 years: [tex]\( \frac{5700}{5700} = 1 \)[/tex] half-life
- 11,400 years: [tex]\( \frac{11400}{5700} = 2 \)[/tex] half-lives
- 17,100 years: [tex]\( \frac{17100}{5700} = 3 \)[/tex] half-lives
- 22,800 years: [tex]\( \frac{22800}{5700} = 4 \)[/tex] half-lives
- 28,500 years: [tex]\( \frac{28500}{5700} = 5 \)[/tex] half-lives
- 34,200 years: [tex]\( \frac{34200}{5700} = 6 \)[/tex] half-lives
- 39,900 years: [tex]\( \frac{39900}{5700} = 7 \)[/tex] half-lives
- 45,600 years: [tex]\( \frac{45600}{5700} = 8 \)[/tex] half-lives
- 51,300 years: [tex]\( \frac{51300}{5700} = 9 \)[/tex] half-lives
### Step 2: Plot the Data
We will use the converted half-lives as our [tex]\( x \)[/tex]-axis and the given amounts of carbon-14 as our [tex]\( y \)[/tex]-axis. Ensure the [tex]\( x \)[/tex]-axis is labeled "Number of half-lives" and the [tex]\( y \)[/tex]-axis is labeled "Amount of isotope (g)."
### Step 3: Create the Graph
To create a graph, follow these points:
1. Plot the points on a coordinate system.
2. Label the [tex]\( x \)[/tex]-axis as "Number of half-lives".
3. Label the [tex]\( y \)[/tex]-axis as "Amount of isotope (g)".
4. Draw a smooth curve through the points to represent the decay.
Here is the plotted data:
- (0, 10000)
- (1, 5000)
- (2, 2500)
- (3, 1250)
- (4, 625)
- (5, 312)
- (6, 156)
- (7, 78)
- (8, 39)
- (9, 20)
### Step 4: Example Plot
Imagine the graph as depicted below, but you can plot it on graph paper or use graphing software:
[tex]\[ \begin{array}{ccccccccc} \text{Number of half-lives (x)} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \text{Amount of isotope (g) (y)} & 10000 & 5000 & 2500 & 1250 & 625 & 312 & 156 & 78 & 39 & 20 \\ \end{array} \][/tex]
### Example Plot Description:
- The x-axis will show the number of half-lives ranging from 0 to 9.
- The y-axis will show the amount of isotope in grams, ranging from 0 to 10000.
- Each point will be plotted according to the match of half-lives and carbon-14 remaining.
- Connect the points to see the exponential decay pattern of the carbon-14 over time.
Once plotted, you will notice the exponential decay curve, indicating that the amount of carbon-14 decreases by half every 5,700 years.
This visual representation helps in understanding how radioactive decay progresses over multiple half-lives.
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