Question
Question 11
Read each of the following descriptions and identify the quantities that are undergoing exponential decay.
the amount of radioactive material in a 250-gram sample decays by 50% each day
a 220-mg dose of aspirin decreases by 30% every hour
the value of a rare coin initially priced at $0.05 increases by 35% each month
a ball is picked up off the ground, thrown into the air, increases to a height of 10 feet, and then falls back down to the ground at 0 feet
Choose one situation in part A that represents exponential decay. Write an equation to model the relationship between the two quantities.
For the situation from part A that you chose, explain the meaning of x, y, a, and b in the context of the situation described.
For the situation from part A that you chose, create a table of values and use the table of values to graph the relationship between the two quantities.
Read each of the following descriptions and identify the quantities that are undergoing exponential decay.
the amount of radioactive material in a 250-gram sample decays by 50% each day
a 220-mg dose of aspirin decreases by 30% every hour
the value of a rare coin initially priced at $0.05 increases by 35% each month
a ball is picked up off the ground, thrown into the air, increases to a height of 10 feet, and then falls back down to the ground at 0 feet
Choose one situation in part A that represents exponential decay. Write an equation to model the relationship between the two quantities.
For the situation from part A that you chose, explain the meaning of x, y, a, and b in the context of the situation described.
For the situation from part A that you chose, create a table of values and use the table of values to graph the relationship between the two quantities.
Answers
GPT-4o mini
To identify the quantities that are undergoing exponential decay from the descriptions provided:
1. The amount of radioactive material in a 250-gram sample decays by 50% each day. **(Exponential decay)**
2. A 220-mg dose of aspirin decreases by 30% every hour. **(Exponential decay)**
3. The value of a rare coin initially priced at $0.05 increases by 35% each month. **(Not exponential decay; this is exponential growth)**
4. A ball is picked up off the ground, thrown into the air, increases to a height of 10 feet, and then falls back down to the ground at 0 feet. **(Not exponential decay; this describes a physical motion scenario)**
### Chosen Situation
**Situation:** The amount of radioactive material in a 250-gram sample decays by 50% each day.
### Equation Modeling Exponential Decay
The general formula for exponential decay can be expressed as:
\[ y = a \cdot b^x \]
where:
- \( y \) represents the remaining quantity after time \( x \),
- \( a \) is the initial amount,
- \( b \) is the decay factor (where \( 0 < b < 1 \)),
- \( x \) is the number of time periods (days in this case).
For this situation:
1. **Initial amount (a)**: 250 grams
2. **Decay factor (b)**: Since the sample decays by 50% each day, the remaining amount is 50%, or \( 0.5 \). So, \( b = 0.5 \).
3. **Time (x)**: The number of days.
Thus, the equation becomes:
\[ y = 250 \cdot (0.5)^x \]
### Explanation of Variables
- \( x \): Represents the number of days elapsed since the start of the decay (day 0).
- \( y \): Represents the remaining amount of radioactive material after \( x \) days.
- \( a \): The initial quantity of radioactive material (250 grams).
- \( b \): The decay factor (0.5), which indicates that each day, the amount of material is reduced to 50% of what it was the previous day.
### Table of Values
Let's create a table for the first 5 days:
| Day (x) | Remaining Amount (y) = 250 * (0.5)^x |
|---------|-------------------------------------|
| 0 | 250 * (0.5)^0 = 250 grams |
| 1 | 250 * (0.5)^1 = 125 grams |
| 2 | 250 * (0.5)^2 = 62.5 grams |
| 3 | 250 * (0.5)^3 = 31.25 grams |
| 4 | 250 * (0.5)^4 = 15.625 grams |
| 5 | 250 * (0.5)^5 = 7.8125 grams |
### Graphing the Relationship
To graph the relationship, you can plot the values from the table on a coordinate system where the x-axis represents the number of days (x), and the y-axis represents the remaining mass of the radioactive material (y).
- Points to plot: (0, 250), (1, 125), (2, 62.5), (3, 31.25), (4, 15.625), (5, 7.8125).
The graph will show a continuous curve that decreases rapidly at first and then flattens out, illustrating the nature of exponential decay.
1. The amount of radioactive material in a 250-gram sample decays by 50% each day. **(Exponential decay)**
2. A 220-mg dose of aspirin decreases by 30% every hour. **(Exponential decay)**
3. The value of a rare coin initially priced at $0.05 increases by 35% each month. **(Not exponential decay; this is exponential growth)**
4. A ball is picked up off the ground, thrown into the air, increases to a height of 10 feet, and then falls back down to the ground at 0 feet. **(Not exponential decay; this describes a physical motion scenario)**
### Chosen Situation
**Situation:** The amount of radioactive material in a 250-gram sample decays by 50% each day.
### Equation Modeling Exponential Decay
The general formula for exponential decay can be expressed as:
\[ y = a \cdot b^x \]
where:
- \( y \) represents the remaining quantity after time \( x \),
- \( a \) is the initial amount,
- \( b \) is the decay factor (where \( 0 < b < 1 \)),
- \( x \) is the number of time periods (days in this case).
For this situation:
1. **Initial amount (a)**: 250 grams
2. **Decay factor (b)**: Since the sample decays by 50% each day, the remaining amount is 50%, or \( 0.5 \). So, \( b = 0.5 \).
3. **Time (x)**: The number of days.
Thus, the equation becomes:
\[ y = 250 \cdot (0.5)^x \]
### Explanation of Variables
- \( x \): Represents the number of days elapsed since the start of the decay (day 0).
- \( y \): Represents the remaining amount of radioactive material after \( x \) days.
- \( a \): The initial quantity of radioactive material (250 grams).
- \( b \): The decay factor (0.5), which indicates that each day, the amount of material is reduced to 50% of what it was the previous day.
### Table of Values
Let's create a table for the first 5 days:
| Day (x) | Remaining Amount (y) = 250 * (0.5)^x |
|---------|-------------------------------------|
| 0 | 250 * (0.5)^0 = 250 grams |
| 1 | 250 * (0.5)^1 = 125 grams |
| 2 | 250 * (0.5)^2 = 62.5 grams |
| 3 | 250 * (0.5)^3 = 31.25 grams |
| 4 | 250 * (0.5)^4 = 15.625 grams |
| 5 | 250 * (0.5)^5 = 7.8125 grams |
### Graphing the Relationship
To graph the relationship, you can plot the values from the table on a coordinate system where the x-axis represents the number of days (x), and the y-axis represents the remaining mass of the radioactive material (y).
- Points to plot: (0, 250), (1, 125), (2, 62.5), (3, 31.25), (4, 15.625), (5, 7.8125).
The graph will show a continuous curve that decreases rapidly at first and then flattens out, illustrating the nature of exponential decay.