Question

The crop period is divided into four stages, initial, grand growth, mid-season, and late-season, with each stage spanning 25 days. The crop coefficient (\(K_c\)) varies linearly from 0.4 at the initial stage to 1.2 at the mid-season stage. The reference crop evapotranspiration on the 40th day is 8 mm. The corresponding actual crop evapotranspiration (in mm) is ________.

Hint:

Find where day 40 falls among the four growth stages, then interpolate the crop coefficient before multiplying by the reference evapotranspiration.

Answer 1

Correct Answer: 7.04

Step 1: Identify the crop stage for day 40.
Each of the four crop stages spans 25 days, so the stages run from day 0 to 25 (initial), day 25 to 50 (grand growth), day 50 to 75 (mid-season), and day 75 to 100 (late-season).
Day 40 falls inside the grand growth stage, between day 25 and day 50.

Step 2: Set up the linear variation of \(K_c\).
The crop coefficient \(K_c\) is 0.4 at the start of the initial stage (day 0) and rises linearly to 1.2 by the start of the mid-season stage (day 50).
Since the initial stage holds \(K_c\) at 0.4 through day 25, the full rise from 0.4 to 1.2 happens across the grand growth stage, from day 25 to day 50.
So \(K_c\) at any day \(d\) inside grand growth is:
\[ K_c(d) = 0.4 + (1.2 - 0.4) \times \frac{d - 25}{50 - 25} \]

Step 3: Find \(K_c\) on day 40.
Put \(d = 40\) in the expression above.
\[ K_c(40) = 0.4 + 0.8 \times \frac{40 - 25}{25} = 0.4 + 0.8 \times \frac{15}{25} = 0.4 + 0.8 \times 0.6 = 0.4 + 0.48 = 0.88 \]

Step 4: Get the actual evapotranspiration.
Actual crop evapotranspiration uses \(ET_c = K_c \times ET_o\), where \(ET_o\) is the reference evapotranspiration.
Here \(ET_o = 8\) mm on day 40, so:
\[ ET_c = 0.88 \times 8 = 7.04 \text{ mm} \]

Final Answer:
The actual crop evapotranspiration on day 40 works out to 7.04 mm, matching the given key range of 6.80 to 7.10 mm. \[ \boxed{ET_c = 7.04 \text{ mm}} \]