Question

Ram Lakhan has a field in the form of a circle. The cost of ploughing the field at the rate of Rs. $2.24$ per ${{m}^{2}}$ is Rs. $45056$. Find the cost of fencing the field at the rate of Rs. $12.60$ per meter.

Answer 1

Hint: Use the formula of area of circle and perimeter of circle. Use representation ${{m}^{2}}$ and $m$ square meter and meter for area and perimeter respectively. Use the units of the area and perimeter of any two dimensional figure to find the cost of fencing the field.

Complete step-by-step answer:
We know that the area of a circle is $\pi {r^2}$.
And the perimeter or the circumference of a circle is .
Here terms are defined in the following ways:
$r = $ Radius of circle and $\pi = 3.14 \approx \dfrac{{22}}{7}$.
Now, from the given information in the question:
Rate of ploughing $ = {\text{Rs}}{\text{. }}2.24{\text{ per }}{{\text{m}}^2}$.
Cost of ploughing\[ = {\text{Rs}}{\text{. }}45056\]
We can say that the area ploughed for ${\text{Rs}}{\text{. }}2.24$ is $1{\text{ }}{{\text{m}}^2}$
Then the area ploughed for ${\text{Rs}}{\text{. }}45056$ can be calculated as:
\[
   \Rightarrow {\text{Area }} = \dfrac{{45056}}{{2.24}} \\
   \Rightarrow {\text{Area }} = 20114.28{\text{ m}} \\
 \]
Therefore the area of the circular field $ = 20114.28{\text{ }}{{\text{m}}^2}$.
Applying formula of area of circle, we get
$ \Rightarrow \pi {r^2} = 20114.28$, where r is the radius of the field.
Simplifying it further we’ll get:
$
   \Rightarrow {r^2} = \dfrac{{20114.28}}{\pi } \\
   \Rightarrow {r^2} = 20114.28 \times \dfrac{7}{{22}} \\
   \Rightarrow {r^2} = 6400 \\
 $
Thus we have the radius of field as:
$
   \Rightarrow r = \sqrt {6400} \\
   \Rightarrow r = 80 \\
 $
Radius of field $ = 80{\text{ m}}$.
Now we have to find cost of fencing the field if the rate of fencing is ${\text{Rs}}{\text{. }}12.60{\text{ per m}}$.
Here per $m$ denotes that we have to use the perimeter formula for circle and field is in circular form
Therefore, perimeter of circle $ = 2\pi r$, where \[r = \]radius of field$ = 80{\text{ m}}$.
Putting these values to get the perimeter of field:
Perimeter of field $ = 2 \times \dfrac{{22}}{7} \times 80$
Calculating further, we get:
Perimeter of field $ = 502.86m$
Now, given cost of fencing of field for $1{\text{ m}} = {\text{Rs}}{\text{. }}12.60$
Then total cost for fencing it will be:
 $
   \Rightarrow {\text{ Cost}} = 502.68 \times 12.60 \\
   \Rightarrow {\text{ Cost}} = {\text{Rs}}{\text{. }}6336 \\
 $
Hence the cost of fencing the field at the rate of ${\text{Rs}}{\text{. }}12.60$ per meter is ${\text{Rs}}{\text{. }}6336$.

Note: In such a type of question we just have to observe the units. Which means if meter square is given then use the formula for area and if meter is given then use perimeter or circumference. We always take length as a positive quantity, here radius is taken positive. We have to use a unitary method to find the cost of the calculated area.