Question

A circular flower bed is surrounded by a path 4m wide. The diameter of the flower bed is 66m. What is the area of the path? (\[\pi = 3.14\]).

Answer 1

Hint: Here we will find the area of the path by subtracting the area of the smaller circle whose diameter is 66m from the area of the larger circle.
Here we will first find the radius of both the smaller and the larger circle and then apply the formula of area of circle for each of the circles and then finally subtract the area to get the desired answer.

Complete step-by-step answer:

It is given that the diameter of the flower bed is 66m.
Let the inner circle represent the flower bed.
Hence we will first find the radius of the inner circle.
Now we know that the radius is half of the diameter of the circle.
Let \[{r_1}\] be the radius of the inner circle.
Hence,
$\Rightarrow$\[{r_1} = \dfrac{{66}}{2}m\]
Solving it we get:-
$\Rightarrow$\[{r_1} = 33m\]
Now we know that the radius of the outer circle is 4 cm more than the radius of the inner circle.
Then if \[{r_2}\] is the radius of the outer circle then,
$\Rightarrow$\[{r_2} = 33 + 4\]
$\Rightarrow$\[ \Rightarrow {r_2} = 37m\]
Now we will calculate the areas of both inner and outer circles.
We know that the area of a circle is given by:-
\[area = \pi {r^2}\]
Hence applying this formula for inner circle we get:-
$\Rightarrow$\[area\left( {inner} \right) = \pi {\left( {33} \right)^2}\]
Putting the value \[\pi = 3.14\] and solving it we get:-
$\Rightarrow$\[area\left( {inner} \right) = 3.14 \times 33 \times 33\]
Solving it we get:-
$\Rightarrow$\[area\left( {inner} \right) = 3419.46{m^2}\]
Now we will calculate the area of the outer circle:-
$\Rightarrow$\[area\left( {outer} \right) = \pi {\left( {37} \right)^2}\]
Putting the value \[\pi = 3.14\] and solving it we get:-
$\Rightarrow$\[area\left( {outer} \right) = 3.14 \times 37 \times 37\]
Solving it we get:-
$\Rightarrow$\[area\left( {outer} \right) = 4298.66{m^2}\]
Now in order to get the area of the path we need to subtract the area of the inner circle from the area of the outer circle.
Hence we get:-
\[{\text{area of the path}} = area\left( {outer} \right) - area\left( {inner} \right)\]
Putting the values we get:-
$\Rightarrow$\[{\text{area of the path}} = 4298.66 - 3419.46\]
Solving it further we get:-
$\Rightarrow$\[{\text{area of the path}} = 879.2{m^2}\]

Hence the required area is \[879.2{m^2}\]

Note: Students should note that they can calculate the area of the path directly by using the following
formula:- \[{\text{area of the path}} = \pi \left( {{r_2}^2 - {r_1}^2} \right)\]
Putting the respective values we get:-
\[{\text{area of the path}} = 3.14\left( {{{\left( {37} \right)}^2} - {{\left( {33} \right)}^2}} \right)\]
Solving it further we get:-
\[{\text{area of the path}} = 3.14\left( {280} \right)\]
Simplifying it we get:-
\[{\text{area of the path}} = 879.2{m^2}\]