An irrigation system's development and construction expenses average $1,600 per hectare. If the end users have 15 years to pay back the loan used to finance the construction, a.) what is the contribution to a sinking fund the end-users would have to pay in annual irrigation fees per hectare if the rate of interest on the fund is 7% compounded annually? b.) If the interest on the loan is 12% per annum, and the operation and maintenance expenses, estimated to be $37.00 per hectare per year, were also assumed by the end-users instead of by the government, what would be the total annual irrigation charge per hectare?

### Part (a) - Contribution to a Sinking Fund

To find the annual contribution to the sinking fund, we use the sinking fund formula. This formula determines the annual payment required to accumulate a future value (i.e., the principal of the loan) given a fixed interest rate and time period:

\[
A = \frac{P \times i}{(1 + i)^n - 1}
\]

Where:
- $A$ is the annual contribution to the sinking fund,
- $P$ is the total loan amount per hectare (i.e., \$1,600),
- $i$ is the interest rate per period (i.e., 7% compounded annually or 0.07),
- $n$ is the number of periods (i.e., 15 years).

We can now calculate $A$:

\[
A = \frac{1600 \times 0.07}{(1 + 0.07)^{15} - 1}
\]

Let's compute this.

#### Step-by-Step Calculation for Part (a):

\[
A = \frac{1600 \times 0.07}{(1.07)^{15} - 1}
\]

Now, I will calculate the value of $A$.The annual contribution to the sinking fund required by the end-users is approximately **$63.67** per hectare.

### Part (b) - Total Annual Irrigation Charge

In addition to the sinking fund contribution, the end-users also have to cover the loan interest and operation/maintenance expenses. To calculate the total annual irrigation charge per hectare, we will consider the following:

1. **Loan Interest Payment**: This is calculated using the formula for the annual payment on a loan with compound interest:

   \[
   I = \frac{P \times r}{1 - (1 + r)^{-n}}
   \]

   Where:
   - $ I $ is the annual loan payment,
   - $ P $ is the loan amount (\$1,600),
   - $ r $ is the annual interest rate on the loan (12% or 0.12),
   - $ n $ is the repayment period in years (15 years).

2. **Operation and Maintenance Expenses**: These are directly provided as \$37.00 per hectare per year.

We will now calculate the loan interest payment and then sum all components (sinking fund, loan payment, and O&M expenses).

Let me compute this.The annual loan payment per hectare, considering a 12% interest rate, is approximately **$234.92**. 

Thus, the total annual irrigation charge per hectare, which includes the sinking fund contribution, loan payment, and operation and maintenance expenses, is approximately **$335.59**.

### Summary:
- **(a)** The annual sinking fund contribution is **$63.67** per hectare.
- **(b)** The total annual irrigation charge per hectare is **$335.59**.

Would you like further clarification on any part of the solution?

Here are 5 related questions to explore:
1. How would the annual irrigation charge change if the loan interest rate were 10%?
2. What would be the total cost per hectare if the repayment period was extended to 20 years?
3. How would the sinking fund contribution change if the interest rate were compounded semi-annually?
4. What is the total amount paid over the 15-year period by end-users for one hectare?
5. How do different loan repayment plans (e.g., interest-only loans) impact the total irrigation charge?

**Tip**: Understanding how interest compounds can help you predict costs better over long loan terms.

Explore the detailed solution to calculate the annual irrigation fees for a system costing $1,600 per hectare. Learn how to compute the sinking fund contribution and the total annual charge considering loan repayment with a 12% interest rate and operation expenses of $37.00 per hectare. Suitable for finance students.

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