Question

A rectangular lawn \(60 \mathrm{~m} \times 40 \mathrm{~m}\) has two roads each \(5 \mathrm{~m}\) wide running in the middle of it, one parallel to length and the other parallel to breadth. The cost of gravelling the roads at 80 paise per sq. \(\mathrm{m}\) is : (a) Rs. 380 (b) Rs. 385 (c) Rs. 400 (d) none of these

Short answer

Answer: (a) Rs. 380

Step-by-step solution

Calculate the area of the roads

First, we must determine the lengths of the two roads. The road parallel to the length of the lawn is 60 m and the road parallel to the breadth of the lawn is 40 m. Both roads have a width of 5 m. Now, let's calculate the area of each road. Road 1 (parallel to length): Length = 60 m; Width = 5 m Area of Road 1 = Length × Width = 60 m × 5 m = 300 m² Road 2 (parallel to breadth): Length = 40 m; Width = 5 m Area of Road 2 = Length × Width = 40 m × 5 m = 200 m² However, we have to subtract the area of the intersection of the two-roads since it was calculated twice. Intersection: Length = 5 m; Width = 5 m Area of Intersection = Length × Width = 5 m × 5 m = 25 m² Total area of the roads = Area of Road 1 + Area of Road 2 - Area of Intersection = 300 m² + 200 m² - 25 m² = 475 m²

Calculate the cost of gravelling the roads

Now that we have the total area of the roads, we can calculate the cost of gravelling. The cost of gravelling per sq.m is 80 paise. Total cost of gravelling = Total area of the roads × Cost per sq.m = 475 m² × 80 paise Now, we need to convert the paise to Indian Rupees (Rs.). 1 Rs. = 100 paise Total cost of gravelling = 475 m² × 80 paise × (1 Rs./100 paise) = 475 m² × 0.8 Rs. = Rs. 380

Answer:

The correct answer is (a) Rs. 380

Key concepts

Area calculation

Understanding how to calculate the area is crucial, especially when dealing with shapes like rectangles. The area of a rectangle is simply the product of its length and its width. For example, for a road running parallel to the lawn's length, the area is calculated as:
\[\text{Area} = \text{Length} \times \text{Width}\]This method is used separately for each of the two roads in our problem. However, when the roads cross each other, they form an intersection that has been counted twice. To adjust for this, we calculate the area of the intersection and subtract it from the total area of both roads complete. This ensures an accurate total area measurement.

Cost estimation

Estimating cost effectively is important in budget planning. Once the total area is known, the next step is to calculate the cost to cover that area. In our example, the cost to gravel the roads is provided at 80 paise per square meter. Here's how we convert this to Indian Rupees:

• 1 Rs. equals 100 paise.
• The conversion involves multiplying the total area by the cost per sq.m.
• The result in paise is then converted by dividing by 100 to reflect the cost in Rs.

Paying attention to units is essential because using the incorrect units can lead to errors in the cost calculated.
It's a straightforward multiplication and conversion process that simplifies budgeting when the area and rate are known.

Rectangular lawn

Rectangular lawns, or any rectangular areas, have distinctive properties that influence planning and designing within the space. A rectangle's definition is a shape with opposite sides of equal length. This means any calculations or modifications affect both length and width dimensions.

• In our example, the lawn is 60 meters by 40 meters.
• Adding features, like roads, needs careful calculation of impact on total area.

A well-measured layout allows for efficient use of resources like gravel, and understanding the lawn's dimensions is the foundation of all other calculations.

Intersection of roads

When roads intersect, the area where they cross overlaps. This overlap requires careful adjustment in calculations, as areas are mistakenly counted twice. In the context of a rectangular lawn with two roads:

• Each road's area adds to the total coverage.
• The intersection, a smaller square, consists of the width of each road overlapping the other's path.
• The area of this overlap must be subtracted from the total of combined road areas.

This adjustment avoids overestimation of resources, such as gravel needed, ensuring cost accuracy.
Managing intersections properly is key to precise area measurement and subsequent cost estimation.