Carrying capacity: J-shaped and S-shaped growth

Grade 9 Science Worksheet - Challenge Me

Circle the best answer.

1. 1.

   Ari reads the graph and uses growth = r × N × (K − N) ÷ K with r = 0.5. How many ducks are added in the year after year 3?

   • 60 ducks
   • 120 ducks
   • 30 ducks
   • 15 ducks

2. Read this first

   Three groups of plants with the same rate (model data)

   group	count now (N)	carrying capacity (K)
   A	20	200
   B	100	200
   C	180	200

3. 2.

   Bo reads the table. Each group uses growth = r × N × (K − N) ÷ K with r = 0.5. Which group adds the most this year?

   • Group B, because 100 is half of 200
   • Groups A and C tie, each adding 9
   • Group C, because it has the most plants
   • Group A, because it has the most room left, 180

4. Bacteria in a flask that holds 1000 (model data)

   hour	bacteria
   0	50
   1	100
   2	200
   3	400

5. 3.

   Esme reads the table. The count doubles each hour. In which hour would doubling first pass the flask's carrying capacity of 1000?

   • Hour 5, when the count would be 1600
   • Hour 4, when the count would be 800
   • Hour 20, when the count would be 1050
   • Hour 6, when the count would be 3200
6. 4.

   Luz reads the graph. After a dry year at year 5, the count falls to a new flat level. What explains it?

   • The line must rise again soon, since populations have no limit
   • The carrying capacity is the 10 ducks of the start, so it fell back
   • The carrying capacity is still 120, so the fall is a counting slip
   • The dry year lowered the carrying capacity from 120 to 80, and the count followed

7. Notebook

   Notes on yeast in a test tube (classmate's notebook), 4 lines. Columns: line, entry. Line 1: The yeast start at 10 thousand cells and level off at 80 thousand. Line 2: The carrying capacity is the starting number, 10 thousand. Line 3: Early on, each cell has plenty of room and sugar, so the count climbs quickly. Line 4: Later, growth slows as the sugar is used up.

8. 5.

   Milo reads a classmate's notebook about yeast. Which line has the mistake?

   • Line 4: growth is fastest once the sugar is used up, because the count is highest
   • Line 3: early growth should be slow, since the cells are few and have little room
   • Line 2: the carrying capacity is the level where growth stops, 80 thousand
   • Line 1: the yeast would keep rising past 80 thousand, so the count should not level off

9. Notebook

   Notes on bacteria in a flask (classmate's notebook), 4 lines. Columns: line, entry. Line 1: Start: 20 bacteria in a flask with plenty of food. Line 2: Hour 1: 20 doubled is 40. Line 3: Hour 2: 40 doubled is 80. Line 4: Hour 3: 80 + 40 = 120, adding the same 40 as last time.

10. 6.

    Noah reads a classmate's notebook about bacteria. Which line has the mistake?

    • Line 4: doubling 80 gives 160, not 120
    • Line 3: the count should be 60, adding 20 each hour
    • Line 1: the flask should start with 10 bacteria
    • Line 2: doubling 20 gives 20 + 20 + 20 = 60

11. Notebook

    Notes on logistic growth (classmate's notebook), 4 lines. Columns: line, entry. Line 1: Early on, N is small and K − N is large, so growth is almost J-shaped. Line 2: Near K, the fraction (K − N) ÷ K is close to 0, so growth slows. Line 3: So growth is fastest when N reaches K. Line 4: At N = K, the count stays near K.

12. 7.

    Pia reads a classmate's notebook about logistic growth. Which line has the mistake?

    • Line 4: at N = K the count keeps climbing
    • Line 2: near K the fraction is close to 1, so growth speeds up
    • Line 3: at N = K the fraction is 0, so growth is 0
    • Line 1: early growth is slow and straight, not J-shaped
13. 8.

    Ren reads the graph. Which claim goes with the right reason?

    • Claim: the frogs will keep rising past 90. Reason: the count is still rising at the end, from 89 to 90
    • Claim: the carrying capacity is 3. Reason: that is the first count on the line, at the start of the graph
    • Claim: growth is fastest near the carrying capacity. Reason: the count is largest there, 89, so the most are added
    • Claim: growth has slowed near the carrying capacity. Reason: from year 9 to year 10 the gain is 2, against 17 at the steepest

14. Two ponds of duckweed (model data)

    pond	plants at the start	plants at the flat part	weeks to reach half of the flat part
    A	20	400	6
    B	20	250	3

15. 9.

    Tala reads the table. Which claim goes with the right reason?

    • Claim: the ponds share a carrying capacity. Reason: both hold the same plant
    • Claim: pond B has the higher carrying capacity. Reason: it reached half of its flat part sooner, in 3 weeks
    • Claim: the ponds share a carrying capacity. Reason: each began with 20 plants
    • Claim: pond A has the higher carrying capacity. Reason: it levels off at 400, against 250

16. Bacteria in a flask with plenty of food (model data)

    hour	bacteria
    0	3
    1	6
    2	12
    3	24

17. 10.

    Zee reads the table. The count keeps doubling. Which claim goes with the right reason?

    • Claim: at hour 6 there are 18 bacteria. Reason: 3 times 6 hours
    • Claim: at hour 6 there are 96 bacteria. Reason: 3 is doubled 5 times, 3 × 32
    • Claim: at hour 6 there are 21 bacteria. Reason: 3 is added 6 times, 3 + 6 × 3
    • Claim: at hour 6 there are 192 bacteria. Reason: 3 is doubled 6 times, 3 × 64