To determine the design discharge for the water supply system in the specified rural village, we need to follow a step-by-step approach to estimate the future population and the corresponding water demand. Let's go through these steps in detail:
### Step 1: Calculate the Intervals Between the Given Years
We first find the differences between the consecutive years provided:
[tex]\[
\begin{align*}
\text{Interval}_1 &= 2037 - 2027 = 10 \, \text{years} \\
\text{Interval}_2 &= 2047 - 2037 = 10 \, \text{years} \\
\text{Interval}_3 &= 2057 - 2047 = 10 \, \text{years} \\
\text{Interval}_4 &= 2067 - 2057 = 10 \, \text{years} \\
\end{align*}
\][/tex]
The intervals between years are uniform, each being 10 years.
### Step 2: Calculate the Growth Rates Between Each Interval
Next, we compute the growth rates for the population between each given interval:
[tex]\[
\begin{align*}
\text{Growth Rate}_1 &= \frac{1.14 - 0.85}{10} = 0.029 \, \text{(thousands per year)} \\
\text{Growth Rate}_2 &= \frac{1.32 - 1.14}{10} = 0.018 \, \text{(thousands per year)} \\
\text{Growth Rate}_3 &= \frac{2.16 - 1.32}{10} = 0.084 \, \text{(thousands per year)} \\
\text{Growth Rate}_4 &= \frac{2.96 - 2.16}{10} = 0.080 \, \text{(thousands per year)} \\
\end{align*}
\][/tex]
### Step 3: Calculate the Average Growth Rate
To estimate future population accurately, we calculate the average growth rate over the given periods:
[tex]\[
\text{Average Growth Rate} = \frac{0.029 + 0.018 + 0.084 + 0.080}{4} = 0.05275 \, \text{(thousands per year)}
\][/tex]
### Step 4: Estimate the Population at the End of the Design Period
We now project the population for the year after the design period ends. The last recorded population is for the year [tex]\(2067\)[/tex], and the design period is [tex]\(21\)[/tex] years. Thus, we need to estimate the population for the year [tex]\(2088\)[/tex]:
[tex]\[
\text{Population in 2088} = \text{Population in 2067} + (\text{Average Growth Rate} \times \text{Design period})
\][/tex]
[tex]\[
\text{Population in 2088} = 2.96 + (0.05275 \times 21) = 4.06775 \, \text{(thousands)}
\][/tex]
### Step 5: Determine the Design Discharge
Finally, we assume that each person in the population requires a discharge amount for water supply. Given that the discharge required per person per day is [tex]\(0.1\)[/tex] cubic meters:
[tex]\[
\text{Design Discharge} = \text{Estimated Population} \times 1000 \times \text{Discharge per Person per Day}
\][/tex]
[tex]\[
\text{Design Discharge} = 4.06775 \times 1000 \times 0.1 = 406.775 \, \text{cubic meters per day}
\][/tex]
Considering losses, theft, and the health post requirements, we could safely say the design discharge should incorporate a safety margin. However, based on calculations alone:
[tex]\[
\boxed{406.775 \, \text{cubic meters per day}}
\][/tex]
would be the required design discharge for the given rural village for the specified design period.
