Recover Growth, Initial Stock, and Time
牛吃草模型：求生长量、原有量与时间
Two complete scenarios can reveal the hidden daily growth and the amount that was present at the beginning. Then a net rate tells us how long the resource will last—or how many consumers it can support.
Use the four-step resource engine
A growing-resource problem becomes manageable when you always ask the same four questions in the same order.
The model at the emptying time
Each pair describes separate trials with the same positive starting stock and constant growth. Each trial ends exactly when all grass is gone. Every cow eats at the same constant rate. One cow-week is the amount one cow eats in one week; one cow-day is the amount one cow eats in one day.
S + gt = ntS is the starting stock, g is growth per unit of time, n is consumption per unit of time, and t is time. Use weeks with weekly rates, or days with daily rates.
The net-depletion form
S = (n − g)tOnly the part of consumption above the growth rate reduces the starting stock.
Put the four actions in order
Use two scenarios to reveal hidden quantities
Both scenarios contain the same starting stock. Their difference comes only from growth during the extra time.
The green part is the same starting stock in both bars. The gold part is the amount grown during the scenario.
Scenario machine
The machine works when the two scenarios truly describe the same starting resource and the same constant growth rate.
Read the machine
Pasture investigation: 27 cows and 23 cows
The pasture can feed 27 cows for 6 weeks, or 23 cows for 9 weeks. Find the weekly growth and the original grass.
Scenario A
27 × 6 = 162 cow-weeksScenario B
23 × 9 = 207 cow-weeksFind how long a new herd can eat
A pasture feeds 24 cows for 6 weeks or 18 cows for 10 weeks. How long can it feed 19 cows?
Step 1 · Growth and start
g = (18 × 10 − 24 × 6) ÷ (10 − 6) = 9S = 24 × 6 − 9 × 6 = 90Step 2 · Net cows
Nine cows are matched by the new grass each week. The other ten cows reduce the original stock.
time = 90 ÷ (19 − 9) = 9 weeksFind the sustainable herd size
A pasture feeds 100 cows for 3 weeks or 50 cows for 8 weeks. What is the largest herd it could support indefinitely under this constant-growth model?
Recover the hidden rates
g = (50 × 8 − 100 × 3) ÷ (8 − 3) = 20S = 100 × 3 − 20 × 3 = 240The pasture grows 20 cow-week units every week.
n > g
The original stock decreases and eventually empties.
n = g
New growth exactly replaces consumption. The starting stock stays unchanged.
n < g
The resource grows. It does not empty under the model.
Now set a target time of 2 weeks
The herd must eat the weekly growth and also remove half of the 240-unit starting stock each week.
n = g + S ÷ t = 20 + 240 ÷ 2 = 140 cowsThe same model works for grass cutters
A growing field can be cut by 17 workers in 30 days or by 19 workers in 24 days. How long would 49 workers need?
Translate the story
Workers cut at equal, constant rates. One worker-day unit is the amount one worker cuts in one day. In each separate trial the entire field is cleared at the stated finishing time.
| Pasture model | Cutting model |
|---|---|
| Cows | Workers |
| Grass growth | New grass growing |
| Grass eaten | Grass cut |
| Original grass | Grass present at the start |
Calculate
g = (17 × 30 − 19 × 24) ÷ (30 − 24) = 9S = 17 × 30 − 9 × 30 = 240time = 240 ÷ (49 − 9) = 6 daysWork backward for time or consumers
Once you know the starting stock and growth rate, the same relationship can find either an unknown time or an unknown number of consumers.
Find time
t = S ÷ (n − g)Use this only when n > g.
Find consumers
n = g + S ÷ tEach consumer uses one unit per day. The consumers must cover the new growth and remove the starting stock in the target time.
Backward-solving laboratory
When the herd changes, split the timeline
A pasture feeds 17 cows for 30 days or 19 cows for 24 days. An unknown herd eats for 6 days, then 4 cows are sold, and the remaining herd finishes the grass in 2 more days. Find the original herd size.
First recover the pasture
g = 9 cow-day units per dayS = 240 cow-day unitsNow split the consumption into two periods.
Trial herd-size slider
Recover a different pasture — independent investigation
Try the eight steps yourself, using a small hint if needed. Check an attempt before opening worked review. Correct all eight to complete this investigation.
In separate trials, a pasture runs out of grass after exactly 8 days with 22 cows or exactly 12 days with 18 cows. Both trials have the same starting grass and constant growth. All cows eat equally. Use cow-day units.
Write quantities in one consistent unit. Enter decimals or fractions such as 30/7; a decimal within 0.001 is accepted.
Hint
Multiply rate by time.
Worked solution
22 × 8 = 176.
Hint
Compare totals only after using the same unit.
Worked solution
18 × 12 = 216.
Hint
Use the two trials and cancel their identical starting stock.
Worked solution
(216 − 176) ÷ (12 − 8) = 10.
Hint
Track the extra time or remove all growth from total use.
Worked solution
176 − 8 × 10 = 96; also 216 − 12 × 10 = 96.
Hint
Subtract daily growth from the 26 cows’ daily use.
Worked solution
26 − 10 = 16.
Hint
Divide starting stock by the net daily depletion found in Step 5.
Worked solution
96 ÷ 16 = 6.
Hint
Find how much starting grass must be removed each day, then add the daily growth.
Worked solution
10 cows match growth; 96 ÷ 4 = 24 more remove the start. Total 34.
Hint
The largest sustainable herd eats exactly as much as grows each day.
Worked solution
Ten cows exactly match growth. Any larger herd reduces the starting stock.
Exit ticket
Answer all five questions correctly. Complete the other core missions and the independent investigation to earn the certificate.
Lesson checkpoints completed
Growth–Stock–Time Detective
You completed the checkpoints on growth, starting stock, net depletion and changing herds. Revisit any steps for which you needed solution help.
Lesson 8.2 completed
Teaching notes
Interactive scenario bars, sustainable-herd graph, backward-solving laboratory, trial herd slider, feedback, and assessments are added teaching scaffolds.