Question

environment has less space and fewer resources available for additional population growth. the size of a population growing logistically over time is modeled by an s - shaped growth curve that levels off as population size approaches the carrying capacity. bacterial population chart of bacterial population growth with time on x - axis (hours) and population size (in billions) on y - axis order the time periods. start with the period during which the population had the lowest growth rate. end with the period during which the population had the highest growth rate. lowest growth rate options: between hours 16 and 18, between hours 12 and 14, between hours 2 and 4, between hours 14 and 16, between hours 18 and 20

Explanation:

Step1: Understand Growth Rate from Graph

The growth rate of a population in a logistic growth curve (S - shaped) is related to the slope of the curve at a given time interval. A flatter slope means a lower growth rate, and a steeper slope means a higher growth rate.

Step2: Analyze Each Time Interval

- **Between hours 2 and 4**: The curve is almost flat (population size changes very little), so growth rate is very low.
- **Between hours 12 and 14**: The curve is still relatively flat (just starting to rise), growth rate is low but higher than 2 - 4? Wait, no, looking at the graph, from 0 - 14, the curve is flat, then starts to rise. Wait, re - examining:
- Between 2 - 4: The population is almost constant (flat line), growth rate ≈ 0.
- Between 12 - 14: The population is just starting to increase, slope is small.
- Between 14 - 16: The slope is increasing, steeper than 12 - 14.
- Between 16 - 18: Slope is steeper than 14 - 16? Wait, no, the steepest part is around 14 - 18? Wait, the curve has a period of rapid growth (the steep part) and then levels off.
- Wait, the correct way: The growth rate (slope) is lowest when the curve is flat (early stages, before the exponential growth phase) and highest during the steepest part of the curve (the exponential growth phase).

- So, the order from lowest to highest growth rate:
- Between hours 2 and 4 (almost flat, lowest growth)
- Between hours 12 and 14 (still flat, low growth)
- Between hours 14 and 16 (starting to rise, moderate growth)
- Between hours 16 and 18 (steeper, higher growth)
- Between hours 18 and 20? Wait, no, after the steep part, the curve levels off. Wait, maybe I misread the intervals. Wait the intervals given are: between 2 - 4, 12 - 14, 14 - 16, 16 - 18, 18 - 20? Wait the original options: "between hours 16 and 18", "between hours 12 and 14", "between hours 2 and 4", "between hours 14 and 16", "between hours 18 and 20".

- Let's re - order based on slope:
- Lowest growth rate: Between hours 2 and 4 (flat, almost no growth)
- Next: Between hours 12 and 14 (just starting to grow, small slope)
- Next: Between hours 14 and 16 (slope increasing)
- Next: Between hours 16 and 18 (steeper slope)
- Highest growth rate: Wait, no, when does the slope peak? In logistic growth, the maximum growth rate is at the inflection point, which is around the middle of the steep part. But from the options, let's check the slopes:

- Between 2 - 4: slope ≈ 0 (lowest)
- Between 12 - 14: slope is small (next lowest)
- Between 14 - 16: slope is larger than 12 - 14
- Between 16 - 18: slope is larger than 14 - 16
- Between 18 - 20: The curve is starting to level off, slope is less than 16 - 18? Wait, no, the graph shows that after a certain point (around 20), the curve levels off. Wait, maybe the options have a typo, but based on the given options, the order from lowest to highest growth rate (based on slope) is:

1. between hours 2 and 4 (lowest, flat)
2. between hours 12 and 14 (next, still flat)
3. between hours 14 and 16 (starting to rise)
4. between hours 16 and 18 (steeper rise)
5. Wait, but the last option is between 18 and 20. Wait, maybe I made a mistake. Let's look at the graph again. The population size (y - axis) is in billions. From 0 - 14 hours, the population is almost 0 (flat line). Then from 14 - 20, it rises rapidly, and then levels off after 20. So the slope (growth rate) is:

- 2 - 4: slope ≈ 0 (lowest)
- 12 - 14: slope ≈ 0 (next lowest, since still flat)
- 14 - 16: slope is positive, starting to increase
- 16 - 18: slope is steeper than 14 - 16
- 18 - 20: Wait, no, after 18 - 20, the curve is still rising but starting to level off? Wait, the graph shows that after around 20, the population becomes constant. So between 18 - 20, the slope is less than between 16 - 18? No, maybe the intervals are:

Wait the correct order of growth rate (from lowest to highest) based on the slope of the logistic curve:

The growth rate is lowest when the population is in the lag phase (flat part, before exponential growth) and highest in the exponential growth phase (steepest part of the curve).

So:

- Lowest: between hours 2 and 4 (lag phase, flat)
- Next: between hours 12 and 14 (still lag phase, flat)
- Next: between hours 14 and 16 (start of exponential growth, slope increasing)
- Next: between hours 16 and 18 (exponential growth, steeper slope)
- Highest: Wait, but the option is between 18 and 20? No, maybe the option "between hours 16 and 18" and "between hours 18 and 20" - actually, the steepest part is around 14 - 18, and then it starts to level off. So maybe the intended order is:

1. between hours 2 and 4 (lowest)
2. between hours 12 and 14 (next lowest)
3. between hours 14 and 16 (next)
4. between hours 16 and 18 (next)
5. Wait, no, the problem says "start with the period during which the population had the lowest growth rate. End with the period during which the population had the highest growth rate."

So re - evaluating:

- Between 2 - 4: growth rate is very low (flat)
- Between 12 - 14: growth rate is low (still flat)
- Between 14 - 16: growth rate is moderate (slope increasing)
- Between 16 - 18: growth rate is high (steeper slope)
- Wait, but the last option is between 18 - 20. Maybe there is a mistake in the options, but based on the graph, the order from lowest to highest growth rate (slope) is:

1. between hours 2 and 4
2. between hours 12 and 14
3. between hours 14 and 16
4. between hours 16 and 18
5. Wait, no, the highest growth rate should be in the steepest part. Let's assume that the intervals are:

The correct order (from lowest to highest growth rate) is:

- Lowest: between hours 2 and 4
- Next: between hours 12 and 14
- Next: between hours 14 and 16
- Next: between hours 16 and 18
- Highest: Wait, maybe the option "between hours 16 and 18" is part of the exponential phase. But the options given are: between 2 - 4, 12 - 14, 14 - 16, 16 - 18, 18 - 20.

So the order from lowest to highest growth rate is:

1. between hours 2 and 4 (lowest)
2. between hours 12 and 14 (next lowest)
3. between hours 14 and 16 (next)
4. between hours 16 and 18 (next)
5. Wait, no, maybe the highest is between 14 - 16? No, the slope gets steeper as we move from 14 to 18.

Wait, perhaps the problem has a typo, but based on the standard logistic growth curve:

- Lag phase (low growth): 0 - 14 hours (flat)
- Exponential phase (high growth): 14 - 20 hours (steep)
- Stationary phase (no growth): after 20 hours (flat)

So within the lag phase (2 - 4 and 12 - 14), 2 - 4 is as flat as 12 - 14 (since the curve is flat until around 14). Then in the exponential phase (14 - 20), the slope increases from 14 - 16, 16 - 18, 18 - 20? No, actually, the slope is maximum at the inflection point (middle of exponential phase) and then decreases as it approaches carrying capacity.

But given the options, the order from lowest to highest growth rate is:

1. between hours 2 and 4 (lowest, flat)
2. between hours 12 and 14 (next lowest, still flat)
3. between hours 14 and 16 (start of exponential growth, slope increasing)
4. between hours 16 and 18 (exponential growth, steeper slope)
5. Wait, but the last option is between 18 and 20. Maybe the intended highest is between 16 and 18, but let's proceed with the analysis.

Answer:

1. between hours 2 and 4
2. between hours 12 and 14
3. between hours 14 and 16
4. between hours 16 and 18
5. between hours 18 and 20 (Wait, no, actually, the highest growth rate should be in the middle of the exponential phase, but based on the options, if we consider the slope, the order from lowest to highest is: between 2 - 4 (lowest), between 12 - 14 (next), between 14 - 16 (next), between 16 - 18 (next), between 18 - 20 (highest? No, that's incorrect. But maybe the problem considers that the growth rate increases from 14 - 20, so the order is 2 - 4, 12 - 14, 14 - 16, 16 - 18, 18 - 20. But that's not correct for a logistic curve, but given the options, this is the order.)

Wait, maybe I made a mistake. Let's re - list the intervals with their growth rate (slope) from the graph:

• Between 2 - 4: slope ≈ 0 (lowest)
• Between 12 - 14: slope ≈ 0 (next lowest, since the curve is still flat)
• Between 14 - 16: slope is positive, small (next)
• Between 16 - 18: slope is steeper than 14 - 16 (next)
• Between 18 - 20: slope is less steep than 16 - 18 (but the problem says "end with the highest", so maybe the problem has a different interpretation. Maybe the highest is between 14 - 16, but no. Alternatively, maybe the options are mis - ordered, but based on the visual, the correct order from lowest to highest growth rate is:

1. between hours 2 and 4
2. between hours 12 and 14
3. between hours 14 and 16
4. between hours 18 and 20
5. between hours 16 and 18

No, that doesn't make sense. I think the key is that in the logistic curve, the growth rate is lowest when the population is in the lag phase (flat part) and highest in the exponential phase (steepest part). So the lag phase intervals are 2 - 4 and 12 - 14 (both flat), then the exponential phase intervals are 14 - 16, 16 - 18, 18 - 20 (with slope increasing then decreasing). But since the problem asks to start with lowest and end with highest, the order is:

Lowest: between hours 2 and 4

Next: between hours 12 and 14

Next: between hours 14 and 16

Next: between hours 16 and 18

Highest: between hours 18 and 20 (but this is incorrect for a real logistic curve, but given the options, this is the best we can do.)

Wait, maybe the problem considers that the growth rate is the change in population over time. Let's calculate the approximate growth rate (ΔP/Δt) for each interval:

• Between 2 - 4: Δt = 2, ΔP ≈ 0 (since the curve is flat), growth rate ≈ 0/2 = 0
• Between 12 - 14: Δt = 2, ΔP ≈ 0 (flat), growth rate ≈ 0/2 = 0
• Between 14 - 16: Δt = 2, let's say ΔP ≈ 5 (from graph), growth rate ≈ 5/2 = 2.5
• Between 16 - 18: Δt = 2, ΔP ≈ 10 (steeper), growth rate ≈ 10/2 = 5
• Between 18 - 20: Δt = 2, ΔP ≈ 5 (starting to level off), growth rate ≈ 5/2 = 2.5

Wait, this is different. So in this case, the growth rate is 0 for 2 - 4 and 12 - 14, 2.5 for 14 - 16 and 18 - 20, and 5 for 16 - 18. But the problem says "start with the period during which the population had the lowest growth rate. End with the period during which the population had the highest growth rate."

So the lowest growth rate intervals are 2 - 4 and 12 - 14 (both 0), then 14 - 16 and 18 - 20 (2.5), then 16 - 18 (5). But the problem gives five intervals, so we have to order them. Maybe the graph is different, but based on the visual, the correct order from lowest to highest growth rate is:

1. between hours 2 and 4 (lowest)
2. between hours 12 and 14 (next lowest)
3. between hours 14 and 16 (next)
4. between hours 18 and 20 (next)
5. between hours 16 and 18 (highest)

Yes, this makes sense because 16 - 18 has the steepest slope (highest ΔP/Δt), 14 - 16 and 18 - 20 have moderate slopes, and 2 - 4 and 12 - 14 have the lowest (zero) slopes.