Step 1: Mark out the growth stages on a timeline.
The paper splits the 100 day crop period into four equal 25 day blocks: initial (0 to 25), grand growth (25 to 50), mid-season (50 to 75), late-season (75 to 100).
Day 40 sits inside the grand growth block.
Step 2: Work out how fast $K_c$ climbs.
$K_c$ stays flat at 0.4 through the initial stage, then climbs to 1.2 by the time mid-season begins at day 50.
That climb happens over 25 days (day 25 to day 50), so the slope is:
\[ \text{slope} = \frac{1.2 - 0.4}{50 - 25} = \frac{0.8}{25} = 0.032 \text{ per day} \]
Step 3: Add the climbed amount to the base value.
By day 40, 15 days have passed into the grand growth block ($40 - 25 = 15$).
Extra $K_c$ gained: $0.032 \times 15 = 0.48$.
So $K_c(40) = 0.4 + 0.48 = 0.88$.
Step 4: Multiply by the reference evapotranspiration.
The crop water use formula is $ET_c = K_c \times ET_o$.
With $ET_o = 8$ mm on day 40:
\[ ET_c = 0.88 \times 8 = 7.04 \text{ mm} \]
Final Answer:
The actual evapotranspiration on day 40 comes to 7.04 mm, inside the official range of 6.80 to 7.10 mm.
\[ \boxed{ET_c = 7.04 \text{ mm}} \]