Growth and decay modelling

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Formula

Explanation

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Worked solution

1. The problem involves exponential growth of textbook deliveries, with initial quantity N0 = 5 textbooks on day 1, doubling every T = 2 days. The time t is 6 days. The formula N = N0 × 2^(t/T) models exponential growth.
2. Substitute the given values into the formula: N = 5 × 2^(6/2).
3. Simplify the exponent: 6/2 = 3.
4. Calculate 2^3 = 8.
5. Multiply: 5 × 8 = 40.
6. The total number of textbooks delivered after 6 days is 40.
7. Verify by listing daily deliveries: day 1: 5, day 3: 10, day 5: 20, and on day 6 (after 2 days from day 4), the doubling occurs at the end of day 6. Since doubling happens every 2 days, the pattern is: day 1: 5, day 3: 10, day 5: 20, and on day 6, the next doubling is not complete yet. But the model uses continuous exponential growth based on full doubling intervals.
8. The doubling occurs at t = 2, 4, 6 days. At t = 6, the doubling is complete. So the sequence is: t=0 (day 1): 5, t=2 (day 3): 10, t=4 (day 5): 20, t=6 (day 7): 40. But the delivery is on day 6, which is t=5 days from start? Wait: the model uses t as time in days from start, and the first delivery is on day 1, so t=1 corresponds to day 1.
9. Clarify: the doubling occurs every 2 days, starting from day 1. So the doubling times are at t = 1, 3, 5, 7, etc. But the formula N = N0 × 2^(t/T) assumes continuous exponential growth, not discrete batch deliveries. The problem states: 'delivered in batches that double in quantity every 2 days, starting with 5 textbooks on day 1'. This implies that the batch size doubles every 2 days, not that the cumulative total doubles every 2 days.
10. Re-evaluate: the first batch is 5 on day 1. After 2 days (day 3), the next batch is 10. After another 2 days (day 5), the next batch is 20. On day 6, no new batch is delivered because the next doubling is on day 7. So only three batches: 5 (day 1), 10 (day 3), 20 (day 5). Total delivered by day 6 is 5 + 10 + 20 = 35.
11. The exponential model N = N0 × 2^(t/T) gives the cumulative total at time t. But here, the deliveries are discrete batches, not continuous. The model is intended to represent the cumulative total, not batch size. The problem says: 'the textbooks are delivered in batches that double in quantity every 2 days'. So the batch size doubles, not the cumulative total. Therefore, the cumulative total is the sum of the batches: 5 + 10 + 20 = 35.
12. The correct total number of textbooks delivered after 6 days is 35. The exponential model was misapplied because it gives the cumulative total assuming continuous growth, but here the batch sizes double, so the cumulative total is the sum of geometric series: 5 + 10 + 20 = 35.

Reference answer: 35

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