Runner-bean growth: arithmetic vs geometric model

Full worked solution

1. (i) Model X (linear) predicted length

   Model X is linear growth, so the daily increase is constant and equal to the growth on the first day. First, calculate the fixed daily growth amount: subtract the initial length from the length after 1 day:

   $$\text{Daily growth} = 20.6 - 20 = 0.6 \text{ cm per day}$$

   Total growth after 60 days is the number of days multiplied by the daily growth, added to the initial length:

   $$L_{\text{X}} = 20 + (60 \times 0.6) = 20 + 36 = 56 \text{ cm}$$

   Therefore the predicted length under Model X is 56 cm.

   Note.

   • Linear growth uses $L=L_0+nd$, where $d$ is daily fixed gain.
   • "Exactly 60 days after initial measurement" means 60 full growth periods, no adjustment to day count.

2. (ii) Model Y (exponential) predicted length

   Model Y is exponential growth, so the daily percentage increase is constant. First, find the daily growth factor by dividing the day-1 length by the initial length:

   $$\text{Daily growth factor} = \frac{20.6}{20} = 1.03 \quad (\text{equivalent to } 3\% \text{ daily growth})$$

   For 60 days, the length is the initial length multiplied by the growth factor raised to the power of 60:

   $$L_{\text{Y}} = 20 \times (1.03)^{60}$$

   Evaluate $(1.03{)}^{60}\approx 5.8916$, so:

   $$L_{\text{Y}} = 20 \times 5.8916 \approx 117.83 \approx 118 \text{ cm}$$

   Therefore the predicted length under Model Y, to the nearest whole cm, is 118 cm.

   Note.

   • Exponential growth uses $L=L_0\times r^n$, where $r$ is the daily multiplier.
   • Rounding only happens at the final step, not when calculating the growth factor or $1.03^{60}$.

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