Math Problem Statement

A rectangular grassy lawn measuring 40m by 25m is to be surrounded externally by a path which is 2m wide. Calculate the cost of levelling the path at the rate of ` 8.25 per square metre.

Solution

We are given that a rectangular grassy lawn has dimensions 40 meters by 25 meters, and it is to be surrounded by a 2-meter-wide path. The task is to find the cost of levelling the path, which is calculated at a rate of 8.25 per square meter.

Step 1: Calculate the area of the lawn.

The area of the rectangular lawn is:

Area of Lawn=Length×Width=40m×25m=1000m2\text{Area of Lawn} = \text{Length} \times \text{Width} = 40 \, \text{m} \times 25 \, \text{m} = 1000 \, \text{m}^2

Step 2: Calculate the total area of the lawn and the path.

The path is 2 meters wide and surrounds the lawn externally. Therefore, the overall dimensions (including the path) will be:

Total Length=40m+2×2m=40m+4m=44m\text{Total Length} = 40 \, \text{m} + 2 \times 2 \, \text{m} = 40 \, \text{m} + 4 \, \text{m} = 44 \, \text{m} Total Width=25m+2×2m=25m+4m=29m\text{Total Width} = 25 \, \text{m} + 2 \times 2 \, \text{m} = 25 \, \text{m} + 4 \, \text{m} = 29 \, \text{m}

Now, calculate the total area (lawn + path):

Total Area=Total Length×Total Width=44m×29m=1276m2\text{Total Area} = \text{Total Length} \times \text{Total Width} = 44 \, \text{m} \times 29 \, \text{m} = 1276 \, \text{m}^2

Step 3: Calculate the area of the path.

The area of the path is the difference between the total area and the area of the lawn:

Area of Path=Total AreaArea of Lawn=1276m21000m2=276m2\text{Area of Path} = \text{Total Area} - \text{Area of Lawn} = 1276 \, \text{m}^2 - 1000 \, \text{m}^2 = 276 \, \text{m}^2

Step 4: Calculate the cost of levelling the path.

The cost of levelling the path is calculated by multiplying the area of the path by the cost per square meter:

Cost=Area of Path×Rate per Square Meter=276m2×8.25per m2\text{Cost} = \text{Area of Path} \times \text{Rate per Square Meter} = 276 \, \text{m}^2 \times 8.25 \, \text{per m}^2

Cost=2271(currency)\text{Cost} = 2271 \, \text{(currency)}

Thus, the cost of levelling the path is 2271 currency units.


Would you like further clarification or details on any of the steps? Here are a few related questions:

  1. What would be the area of the path if the width was changed to 3 meters?
  2. How would the cost change if the rate per square meter was increased to 10 currency units?
  3. How do you calculate the area of a rectangle when given its length and width?
  4. How would you approach this problem if the path had varying widths along the lawn?
  5. How can you calculate the cost of levelling for different shapes like circular lawns or irregular plots?

Tip: When dealing with problems involving paths around a rectangular area, always calculate the total area (including the path) and subtract the area of the original object (lawn) to get the area of the path.

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Math Problem Analysis

Mathematical Concepts

Geometry
Area Calculation
Cost Calculation

Formulas

Area of a rectangle = Length × Width
Total Area (including path) = (Length + 2 × Path Width) × (Width + 2 × Path Width)
Area of Path = Total Area - Lawn Area
Cost = Area of Path × Rate per Square Meter

Theorems

Basic Area Theorem for Rectangles

Suitable Grade Level

Grades 6-8