How to find Area between two Concentric Circles

QUESTION: A circular lawn is surrounded by a path of uniform width of 7 meters. The area of the path is 21% that of the lawn. What is the radius of the lawn?

SOLUTION:

Formula of area of a circle is πr²

where r is the radius of a circle.

 

Let r is the radius of a lawn.

Then area of the lawn = πr²

 

Radius of Lawn and path = r + 7

So, Area of the lawn and path = π (r + 7)²

 

Area of the path = Area of the lawn and path – Area of the lawn

= π (r + 7)² – π r²

 

area of the path is 21% that of the lawn (given)

π (r + 7)² – π r² = 21% πr²

 

Simplify

π (r² + 49 + 14r) – π r² = 0.21 π r²

π cancelled on both sides of equal sign

 

r² + 49 + 14 r – r² = 0.21 r²

 

49 + 14 r = 0.21 r²

 

0.21 r² – 14r – 49 = 0

This is a quadratic equation.

 

Here,

a = co-efficient of r² = 0.21

b = co-efficient of r = -14

c = -49

 

Let’s find the values of r by quadratic formula.

 

Put values of a, b and c 

Simplify