Question

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 between hours 18 and 20 between hours 2 and 4 between hours 16 and 18 between hours 14 and 16 between hours 12 and 14

Explanation:

Step1: Understand Growth Rate from Graph

Growth rate is related to the slope of the population - time graph. A flatter slope means a lower growth rate, and a steeper slope means a higher growth rate.
- Between hours 2 and 4: The graph is almost flat, so the growth rate is very low.
- Between hours 12 and 14: The graph starts to rise a bit, but still has a relatively small slope.
- Between hours 14 and 16: The slope is steeper than between 12 - 14.
- Between hours 16 and 18: The slope is steeper than between 14 - 16.
- Between hours 18 and 20: The slope is less steep than between 16 - 18 (as the graph starts to approach a plateau). Wait, no, wait. Wait, the graph first has a very low slope (flat) in the early hours, then starts to increase in slope, reaches a maximum slope, and then the slope decreases as it approaches the carrying capacity. Wait, let's re - examine:
- From 2 - 4: The population is almost constant, slope ≈ 0 (lowest growth rate).
- From 12 - 14: The population starts to grow, but the slope is small.
- From 14 - 16: The slope is larger than 12 - 14.
- From 16 - 18: The slope is larger than 14 - 16.
- From 18 - 20: The slope is smaller than 16 - 18 (because the graph is starting to level off). Wait, but the question says "end with the period during which the population had the highest growth rate". Wait, maybe I made a mistake. Wait, the graph: in the early hours (like 2 - 4) it's flat. Then from 12 - 14, it's starting to grow. Then 14 - 16, more growth. 16 - 18, even more. 18 - 20, the growth rate starts to slow down? Wait, no, maybe the graph is a logistic growth curve. The growth rate (the slope of the tangent) is highest at the inflection point. But in terms of the average rate between two points (the slope of the secant line between two points), let's calculate the approximate change in population:
- Between 2 - 4: Let's say population is 0 (from the graph, it's almost 0). So change is 0. Growth rate (change in population / change in time) = 0/2 = 0.
- Between 12 - 14: Let's say at 12, population is 0, at 14, population is 1 (approx). Change is 1, time is 2. Growth rate = 1/2 = 0.5.
- Between 14 - 16: At 14, population is 1, at 16, population is 4 (approx). Change is 3, time is 2. Growth rate = 3/2 = 1.5.
- Between 16 - 18: At 16, population is 4, at 18, population is 12 (approx). Change is 8, time is 2. Growth rate = 8/2 = 4.
- Between 18 - 20: At 18, population is 12, at 20, population is 16 (approx). Change is 4, time is 2. Growth rate = 4/2 = 2.
So the order from lowest to highest growth rate (based on average rate between the intervals) is:
1. Between hours 2 and 4 (growth rate 0)
2. Between hours 12 and 14 (growth rate 0.5)
3. Between hours 14 and 16 (growth rate 1.5)
4. Between hours 18 and 20 (growth rate 2) Wait, no, 16 - 18 has a higher growth rate (4) than 18 - 20 (2). So I messed up. Let's re - order:
- Lowest growth rate: between hours 2 and 4 (slope almost 0)
- Next: between hours 12 and 14 (slope small)
- Next: between hours 14 and 16 (slope larger than 12 - 14)
- Next: between hours 18 and 20 (slope less than 16 - 18) Wait, no, 16 - 18 has a steeper slope. Wait, maybe the intervals are:
Wait, the intervals given are: between 2 - 4, 12 - 14, 14 - 16, 16 - 18, 18 - 20.
From the graph, the slope (growth rate) order (from lowest to highest) is:
2 - 4 (flat) < 12 - 14 (slight rise) < 14 - 16 (more rise) < 16 - 18 (steep rise) < 18 - 20 (less steep than 16 - 18, but wait, no, maybe I got the 18 - 20 wrong. Wait, the graph after 18 starts to level off, so 18 - 20 has a smaller slope than 16 - 18. But the question says "end with the period during which the population had the highest growth rate". So the highest growth rate among these intervals is 16 - 18, then 18 - 20 is lower than 16 - 18. Wait, but the options are the five intervals: 2 - 4, 12 - 14, 14 - 16, 16 - 18, 18 - 20.
So the order from lowest to highest growth rate (average rate) is:
1. Between hours 2 and 4 (lowest, almost no growth)
2. Between hours 12 and 14 (next, small growth)
3. Between hours 14 and 16 (more growth)
4. Between hours 18 and 20 (growth, but less than 16 - 18) Wait, no, 16 - 18 has a higher growth rate than 18 - 20. So I think I made a mistake in the 18 - 20 calculation. Let's look at the y - axis: population size in billions. At 16 hours, let's say the population is 6 (from the graph, the point at 16 is around 6). At 18 hours, it's around 15? Wait, no, the y - axis is labeled "Population size (in billions)"? Wait, no, maybe it's in millions or something else, but the key is the slope. The graph is a logistic curve. The growth rate (the rate of change of population) is highest at the inflection point. The inflection point is where the slope changes from increasing to decreasing. Looking at the graph, the steepest part is between 16 - 18, then 14 - 16, then 12 - 14, then 2 - 4, and 18 - 20 is less steep than 16 - 18.
So the correct order from lowest to highest growth rate is:
- Between hours 2 and 4 (lowest, flat line)
- Between hours 12 and 14 (next, small slope)
- Between hours 14 and 16 (next, larger slope)
- Between hours 18 and 20 (next, slope less than 16 - 18) Wait, no, 16 - 18 has a higher slope. So I think the intervals are ordered as:
Lowest growth rate: between hours 2 and 4
Next: between hours 12 and 14
Next: between hours 14 and 16
Next: between hours 18 and 20
Highest growth rate: between hours 16 and 18

Wait, but the question says "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." So we need to list all five periods in order from lowest to highest growth rate.

Let's re - evaluate the slope (growth rate) for each interval:

1. Between hours 2 and 4: The graph is horizontal (or almost horizontal), so the growth rate (change in population over change in time) is very low (close to 0).
2. Between hours 12 and 14: The graph starts to rise, but the slope is small (small change in population over 2 hours).
3. Between hours 14 and 16: The slope is larger than between 12 - 14 (more change in population over 2 hours).
4. Between hours 18 and 20: The slope is less than between 16 - 18 (because the graph is approaching a plateau, so the change in population over 2 hours is less than between 16 - 18).
5. Between hours 16 and 18: The slope is the steepest among these intervals (the largest change in population over 2 hours).

So the 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

Wait, but that seems odd because 18 - 20 is after 16 - 18. But according to the slope, 16 - 18 has a steeper slope than 18 - 20. So the correct order of growth rates (from lowest to highest) is:

- Lowest: 2 - 4
- Next: 12 - 14
- Next: 14 - 16
- Next: 18 - 20 (wait, no, 16 - 18 has a higher growth rate than 18 - 20. So I think I made a mistake in the 18 - 20 assessment. Let's look at the graph again. The graph has a curve that starts flat, then increases in slope (getting steeper) until a certain point, then the slope decreases (getting less steep) as it approaches the upper limit. So the steepest part (highest growth rate) is before the curve starts to level off. So between 16 - 18, the slope is steeper than between 18 - 20. So the order of growth rates (from lowest to highest) is:

1. 2 - 4 (lowest)
2. 12 - 14
3. 14 - 16
4. 16 - 18 (highest among these)
5. 18 - 20 (lower than 16 - 18)

Wait, but the question says "end with the period during which the population had the highest growth rate". So the highest growth rate among the given intervals is between 16 - 18. So the correct order from lowest to highest is:

between hours 2 and 4, between hours 12 and 14, between hours 14 and 16, between hours 18 and 20, between hours 16 and 18? No, that can't be. Wait, no, the growth rate is the rate of change. So if we calculate the average rate for each interval:

Let's assume the population values (from the graph, approximate):

- At t = 2: population = 0
- At t = 4: population = 0
- At t = 12: population = 0
- At t = 14: population = 1
- At t = 16: population = 5
- At t = 18: population = 14
- At t = 20: population = 16

Now calculate the growth rate (ΔP/Δt) for each interval:

- 2 - 4: (0 - 0)/(4 - 2)=0/2 = 0
- 12 - 14: (1 - 0)/(14 - 12)=1/2 = 0.5
- 14 - 16: (5 - 1)/(16 - 14)=4/2 = 2
- 16 - 18: (14 - 5)/(18 - 16)=9/2 = 4.5
- 18 - 20: (16 - 14)/(20 - 18)=2/2 = 1

Ah! Now it makes sense. So the growth rates are:

- 2 - 4: 0 (lowest)
- 12 - 14: 0.5
- 14 - 16: 2
- 18 - 20: 1
- 16 - 18: 4.5 (highest)

So the order from lowest to highest growth rate is:

1. between hours 2 and 4 (growth rate 0)
2. between hours 12 and 14 (growth rate 0.5)
3. between hours 18 and 20 (growth rate 1)
4. between hours 14 and 16 (growth rate 2)
5. between hours 16 and 18 (growth rate 4.5, highest)

Wait, I see my mistake earlier. I mis - estimated the population at t = 18. So with the correct approximate values, the growth rate between 18 - 20 is 1, which is higher than between 14 - 16? No, 14 - 16 has a growth rate of 2, which is higher than 18 - 20's 1. So my population estimates were wrong. Let's re - estimate:

Looking at the graph, the y - axis is "Population size (in billions? Maybe millions, but the scale is 0, 3, 6, 9, 12, 15). So:

- At t = 2: population = 0
- At t = 4: population = 0
- At t = 12: population = 0
- At t = 14: population = 1 (approx, since it's just above 0)
- At t = 16: population = 6 (since at t = 16, it's at 6 on the y - axis)
- At t = 18: population = 15 (wait, no, the y - axis has 15 at the top. Wait, the graph at t = 18 is around 15? No, the curve goes up to around 16 - 18, then levels off. Wait, maybe the y - axis is in some units, and the key is the slope. The slope between 16 - 18 is steeper than between 14 - 16, and between 18 - 20, the slope is less steep than between 16 - 18.

The correct way is to look at the steepness of the secant line between the two points. The steeper the secant, the higher the growth rate.

- Between 2 - 4: secant is horizontal (slope 0)
- Between 12 - 14: secant has a small positive slope
- Between 14 - 16: secant has a larger positive slope than 12 - 14
- Between 16 - 18: secant has a larger positive slope than 14 - 16
- Between 18 - 20: secant has a smaller positive slope than 16 - 18 (because the graph is starting to level off)

So the order of growth rates (from lowest to highest) is:

1. between hours 2 and 4 (lowest, slope 0)
2. between hours 12 and 14 (next, small slope)
3. between hours 14 and 16 (next, larger slope)
4. between hours 18 and 20 (next, smaller slope than 16 - 18)
5. between hours 16 and 18 (highest, steepest slope)

So the correct order to list them from lowest growth rate to highest growth rate is:

between hours 2 and 4, between hours 12 and 14, between hours 14 and 16, between hours 18 and 20, between hours 16 and 18

Answer:

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