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The problem states:

A rectangular portion of an airport runway was getting repaired at a rate \( R \) rupees per square unit. However, during the work:

- The length increased by \( 5x \%\).
- The breadth increased by \( 4x \%\).
- The cost per unit area increased by 15%.
  
The overall percentage increase in the total cost is 36.62%. The task is to find the value of \( x \).

Let's break this down step by step:

1. **Initial Cost Formula**: 
   The initial cost of the repair is given by:
   \[
   \text{Initial Cost} = R \times \text{Area} = R \times L \times B
   \]
   where \( L \) is the original length and \( B \) is the original breadth.

2. **New Dimensions After Increase**:
   - The length increases by \( 5x \%\), so the new length is:
     \[
     L_{\text{new}} = L \times \left(1 + \frac{5x}{100}\right)
     \]
   - The breadth increases by \( 4x \%\), so the new breadth is:
     \[
     B_{\text{new}} = B \times \left(1 + \frac{4x}{100}\right)
     \]

3. **New Area**:
   The new area is the product of the new length and new breadth:
   \[
   \text{New Area} = L_{\text{new}} \times B_{\text{new}} = L \times B \times \left(1 + \frac{5x}{100}\right) \times \left(1 + \frac{4x}{100}\right)
   \]

4. **Increase in Rate**:
   The cost per square unit increases by 15%, so the new rate per unit is:
   \[
   R_{\text{new}} = R \times \left(1 + \frac{15}{100}\right) = R \times 1.15
   \]

5. **New Total Cost**:
   The new total cost is:
   \[
   \text{New Cost} = R_{\text{new}} \times \text{New Area} = R \times 1.15 \times L \times B \times \left(1 + \frac{5x}{100}\right) \times \left(1 + \frac{4x}{100}\right)
   \]

6. **Overall Percentage Increase in Cost**:
   The percentage increase in total cost is given as 36.62%. This means:
   \[
   \frac{\text{New Cost} - \text{Initial Cost}}{\text{Initial Cost}} \times 100 = 36.62
   \]
   Simplifying the new cost relative to the initial cost:
   \[
   \text{New Cost} = R \times L \times B \times 1.15 \times \left(1 + \frac{5x}{100}\right) \times \left(1 + \frac{4x}{100}\right)
   \]
   Therefore, the ratio of new cost to initial cost is:
   \[
   \frac{\text{New Cost}}{\text{Initial Cost}} = 1.15 \times \left(1 + \frac{5x}{100}\right) \times \left(1 + \frac{4x}{100}\right)
   \]
   Setting this equal to 1.3662 (since a 36.62% increase corresponds to multiplying by 1.3662):
   \[
   1.15 \times \left(1 + \frac{5x}{100}\right) \times \left(1 + \frac{4x}{100}\right) = 1.3662
   \]

7. **Solving for \( x \)**:
   Now, solving this equation for \( x \).

Let's compute the value of \( x \).The possible values for \( x \) are \( -47 \) and \( 2 \). Since a negative percentage doesn't make sense in this context, the correct value of \( x \) is \( 2 \).

Would you like more details or have any further questions?

Here are five related questions to explore:

1. How would the problem change if the increase in length and breadth were different percentages?
2. What happens to the total cost if the rate \( R \) also changes over time?
3. How can this percentage increase formula be applied to other geometric shapes like circles?
4. What if the area was initially irregular and then approximated as a rectangle?
5. How would the problem change if the increase in repair cost was not uniform across the area?

**Tip:** When dealing with percentage increases involving multiple factors, always remember to compound the percentages correctly.

A rectangular portion of an airport runway was getting repaired for which an estimate was made on the basis of a rate Rs. R per square unit. But while doing the work, the length of the portion got increased by 5x% and the breadth by 4x%. Over and above this, there was an increase in the cost of the repair work to the extent of 15%. If the overall percentage increase in the cost of repair over the estimate is 36.62%, then find the value of x.

Solve a problem involving the calculation of repair costs for an airport runway. Given percentage increases in dimensions and cost per unit, learn how to calculate the overall percentage increase and find the unknown variable x. Ideal for students in Grades 9-12.

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