Question

A square lawn is surrounded by a path 2.5 m wide. If the area of the path is 165 m2, find
the area of the lawn.
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Answer 1

Question:

• A square lawn is surrounded by a path 2.5 m wide. If the area of the path is 165 m2, find  the area of the lawn.

Process:

To find the area of the lawn we will first let the side of the square lawn (ABCD) be x metres and side of the lawn including path will be (x + 25) metres. After that, we will find the area of the lawn(x²) and area of lawn including path(x² + 10x + 25). Then, we will subtract area of the lawn from area of the lawn including path. After subtracting, the answer we got(10x + 25) is the area of the path. Then, we can form an equation as: 10x + 25 = 165. The value of x we got after solving the equation is the side of the lawn. And lastly, we can find the area of the lawn by using formula :

$\boxed{Area \;of \;square = (side^2) sq. \;units}$

Solution:

Let the side of the square lawn be x metre.

Width of the path = 2.5 metre [given]

Then, Side of the lawn including path = {x + 2(2.5)} metres

       = {x + 5} metres

Therefore,

Area of the square lawn (ABCD) = (side²) units

                                     = (x²) metres

Area of the lawn including path (EFGH) = (side²) units

           = {(x+5)²} metres

           = {(x + 5)(x + 5)} metres

           = {x(x + 5) + 5(x + 5)} metres

           = {x² + 5x + 5x + 25} metres

           = {x² + 10x + 25} metres

Area of the path = (Area of EFGH) - (Area of ABCD)

                   = (x² + 10x + 25) - (x²)

                   $= \not{x^2} + 10x + 25 - \not{x^2}$

                    = 10x + 25

But, Area of the path = 165 m²

∴ 10x + 25 = 165 m²

⇒ 10x = 165 - 25

⇒ 10x = 140

$\implies x = \frac{14 \not{0}}{1 \not{0}}$

⇒ x = 14

Hence, side of the lawn = 14 m

Therefore, Area of the lawn = x²

                                    = 14² = 196 m²