Answer :
To determine which graph best represents the relationship between time (in weeks) and the average mass (in milligrams) of a sunfish, we can analyze the given data and the relationship it follows:
1. Analyzing the Growth Data:
The data given is as follows:
- At 0 weeks, the mass is 3.5 milligrams.
- At 5 weeks, the mass is 19.3 milligrams.
- At 10 weeks, the mass is 106.5 milligrams.
- At 15 weeks, the mass is 587.5 milligrams.
- At 20 weeks, the mass is 3240.5 milligrams.
2. Identifying the Pattern:
The mass of the sunfish appears to increase rapidly over time, suggesting an exponential growth pattern. This is consistent with biological growth trends where organisms often experience rapid acceleration in mass over time.
3. Modeling the Growth:
To confirm the exponential growth, we fit the data to an exponential curve of the form [tex]\( M(t) = M_0 \cdot e^{rt} \)[/tex] where:
- [tex]\(M(t)\)[/tex] is the mass at time [tex]\(t\)[/tex],
- [tex]\(M_0\)[/tex] is the initial mass,
- [tex]\(r\)[/tex] is the growth rate,
- [tex]\(t\)[/tex] is the time.
4. Estimates from the Data:
Based on the data analysis, the exponential model provided parameters such that:
- The initial mass ([tex]\(M_0\)[/tex]) is approximately [tex]\(3.5\)[/tex] milligrams,
- The growth rate ([tex]\(r\)[/tex]) is approximately [tex]\(0.342\)[/tex].
5. Creating the Exponential Curve:
Using the exponential model, we can generate the curve over the time span to get an idea of how the mass increases. Below is how we can plot the fitted curve against the discrete data points:
- At time [tex]\(t = 0\)[/tex]: [tex]\(M(0) \approx 3.5\)[/tex]
- At time [tex]\(t = 5\)[/tex]: [tex]\(M(5) \approx 3.5 \cdot e^{0.342 \cdot 5} \approx 19.3\)[/tex]
- At time [tex]\(t = 10\)[/tex]: [tex]\(M(10) \approx 3.5 \cdot e^{0.342 \cdot 10} \approx 106.5\)[/tex]
- At time [tex]\(t = 15\)[/tex]: [tex]\(M(15) \approx 3.5 \cdot e^{0.342 \cdot 15} \approx 587.5\)[/tex]
- At time [tex]\(t = 20\)[/tex]: [tex]\(M(20) \approx 3.5 \cdot e^{0.342 \cdot 20} \approx 3240.5\)[/tex]
6. Choosing the Best Graph:
Based on the inspection of the data points and their fitting curve, the graph that best represents this relationship will show:
- A rapid and steep increase in the sunfish's mass as time progresses.
- The curve starting at a low initial value (approximately 3.5 milligrams at week 0).
- A smooth, continuously increasing curve that represents an exponential climb.
So, the best representation is a graph showing an exponential rise in mass, beginning at around 3.5 milligrams and increasing steeply to just over 3240 milligrams by the 20th week. This graph will not be linear but should have an upward curving trend line that steepens over time.
1. Analyzing the Growth Data:
The data given is as follows:
- At 0 weeks, the mass is 3.5 milligrams.
- At 5 weeks, the mass is 19.3 milligrams.
- At 10 weeks, the mass is 106.5 milligrams.
- At 15 weeks, the mass is 587.5 milligrams.
- At 20 weeks, the mass is 3240.5 milligrams.
2. Identifying the Pattern:
The mass of the sunfish appears to increase rapidly over time, suggesting an exponential growth pattern. This is consistent with biological growth trends where organisms often experience rapid acceleration in mass over time.
3. Modeling the Growth:
To confirm the exponential growth, we fit the data to an exponential curve of the form [tex]\( M(t) = M_0 \cdot e^{rt} \)[/tex] where:
- [tex]\(M(t)\)[/tex] is the mass at time [tex]\(t\)[/tex],
- [tex]\(M_0\)[/tex] is the initial mass,
- [tex]\(r\)[/tex] is the growth rate,
- [tex]\(t\)[/tex] is the time.
4. Estimates from the Data:
Based on the data analysis, the exponential model provided parameters such that:
- The initial mass ([tex]\(M_0\)[/tex]) is approximately [tex]\(3.5\)[/tex] milligrams,
- The growth rate ([tex]\(r\)[/tex]) is approximately [tex]\(0.342\)[/tex].
5. Creating the Exponential Curve:
Using the exponential model, we can generate the curve over the time span to get an idea of how the mass increases. Below is how we can plot the fitted curve against the discrete data points:
- At time [tex]\(t = 0\)[/tex]: [tex]\(M(0) \approx 3.5\)[/tex]
- At time [tex]\(t = 5\)[/tex]: [tex]\(M(5) \approx 3.5 \cdot e^{0.342 \cdot 5} \approx 19.3\)[/tex]
- At time [tex]\(t = 10\)[/tex]: [tex]\(M(10) \approx 3.5 \cdot e^{0.342 \cdot 10} \approx 106.5\)[/tex]
- At time [tex]\(t = 15\)[/tex]: [tex]\(M(15) \approx 3.5 \cdot e^{0.342 \cdot 15} \approx 587.5\)[/tex]
- At time [tex]\(t = 20\)[/tex]: [tex]\(M(20) \approx 3.5 \cdot e^{0.342 \cdot 20} \approx 3240.5\)[/tex]
6. Choosing the Best Graph:
Based on the inspection of the data points and their fitting curve, the graph that best represents this relationship will show:
- A rapid and steep increase in the sunfish's mass as time progresses.
- The curve starting at a low initial value (approximately 3.5 milligrams at week 0).
- A smooth, continuously increasing curve that represents an exponential climb.
So, the best representation is a graph showing an exponential rise in mass, beginning at around 3.5 milligrams and increasing steeply to just over 3240 milligrams by the 20th week. This graph will not be linear but should have an upward curving trend line that steepens over time.
