Question

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

Answer 1

Hint: Here, we need to find the area of the path. To solve the question, we will first find the area of the flower bed and the area of a bigger circle. The radius of the bigger circle is the sum of the radius of the smaller circle and the width of the path. The area of the path will be the difference in the areas of the bigger circle and the flower bed.

Formula Used: We will use the following formulas to solve the question:
1.The radius of a circle, \[r = \dfrac{d}{2}\], where \[r\] is the radius and \[d\] is the diameter of the circle.
2.The area of a circle, \[\pi {r^2}\], where \[r\] is the radius of the circle.

Complete step-by-step answer:
First, we will draw the diagram using the given information.

Here, the yellow region is the circular flower bed with diameter \[AB\] and centre \[O\]. The grey region is the path of width 4 m.
First, we will find the areas of the two concentric circles.
Now, we know that the radius of a circle is half of its diameter, that is \[r = \dfrac{d}{2}\].
Substituting the diameter as 66 m in the formula, we get
\[\begin{array}{l} \Rightarrow r = \dfrac{{66}}{2}\\ \Rightarrow r = 33{\rm{ m}}\end{array}\]
Also, we know that the area of a circle is given by \[\pi {r^2}\], where \[r\] is the radius of the circle.
Substituting \[r = 33{\rm{m}}\], we get
Area of the flower bed \[ = \pi {\left( {33} \right)^2}{{\rm{m}}^2}\]
Simplifying the expression, we get
Area of the flower bed \[ = 1089\pi {\rm{ }}{{\rm{m}}^2}\]
Next, we will calculate the area of the bigger circle.
From the figure, we can see that the radius of the bigger circle is the sum of the radius of the smaller circle and the width of the path.
Therefore, we get
Radius of bigger circle \[ = 33 + 4 = 37{\rm{ m}}\]
Substituting \[r = 37{\rm{m}}\] in the formula for area of a circle, we get
Area of the bigger circle \[ = \pi {\left( {37} \right)^2}{{\rm{m}}^2}\]
Simplifying the expression, we get
Area of the bigger circle \[ = 1369\pi {\rm{ }}{{\rm{m}}^2}\]
Finally, we will find the area of the path of width 4 m.
From the figure, we can observe that the area of the path is equal to the difference in the areas of the bigger circle and the flower bed.
Therefore, we get
\[{\rm{Area\, of\, path}} = {\rm{Area\, of\, bigger\, circle}} - {\rm{Area\, of\, flower\, bed}}\]
Substituting the area of the bigger circle as \[1369\pi {\rm{ }}{{\rm{m}}^2}\] and the area of the flower bed as \[1089\pi {\rm{ }}{{\rm{m}}^2}\], we get
\[\begin{array}{l} \Rightarrow {\rm{Area\, of\, path}} = 1369\pi {\rm{ }}{{\rm{m}}^2} - 1089\pi {\rm{ }}{{\rm{m}}^2}\\ \Rightarrow {\rm{Area\, of\, path}} = 280\pi {\rm{ }}{{\rm{m}}^2}\end{array}\]
Substituting \[\pi = 3.14\], we get
\[\begin{array}{l} \Rightarrow {\rm{Area\, of\, path}} = 280 \times 3.14{\rm{ }}{{\rm{m}}^2}\\ \Rightarrow {\rm{Area\, of\, path}} = 879.2{\rm{ }}{{\rm{m}}^2}\end{array}\]
Therefore, we get the area of the path of width 4 m as \[879.2{\rm{ }}{{\rm{m}}^2}\].

Note: You should remember to subtract the area of the flower bed from the area of the bigger circle to get the area of the path. A common mistake is to assume that the radius of 37 m is to be used to find the area of the circular path. However, that will give the area of the bigger circle and not the area of the path.