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The cost of fencing a circular field at the rate Rs. 24 per metre is Rs. 5280. The field is to be ploughed at the rate of Rs.0.50 per ${m}^2$. Find the cost of ploughing the field. (Take $\pi = \dfrac{{22}}{7}$) 

(A) Rs. 1278 

(B) Rs 1437

(C) Rs 1925

(D) Rs 2870

Last updated date: 13th Jul 2024
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Hint: First find the circumference of the circular field by using the given data. Later find the radius ‘r’ from the circumference formula $2 \pi r$, using this radius find the area of a circular field. To find the cost of ploughing the field multiply the area with the rate of ploughing the field.

Complete step by step answer:

Given: Total cost of fencing the circular field = Rs. 5280

Rate of fencing the circular field = Rs. 24/m

Rate of ploughing the circular field = Rs. $0.5/m^2$

Since, we know that fencing occurs around the circumference and ploughing is done on the area of the circular field.

Therefore, Circumference of the circular field = $\dfrac{{{\text{Total cost of fencing the circular field}}}}{{{\text{Rate of fencing the circular field }}}} $

If 'r' is the radius of the circular field, then its circumference is $2 \pi r$ where $\pi = \dfrac{{22}}{7} $

$ \Rightarrow $ Circumference=$\dfrac{{5280}}{{24}}=2 \pi r$ 

$ \Rightarrow r = \dfrac{{5280}}{{24 \times 2\pi }} = \dfrac{{5280 \times 7}}{{24 \times 2 \times 22}} = 35{\text{ }}m$

So, the radius of the circular field is 35 metres.

Area of the circular field = $\pi {{\text{r}}^2} = \dfrac{{22}}{7} \times {35^2} = 3850{\text{ }}{m^2} $

Now, Total cost of ploughing = (Area of the circular field)$ \times $(Rate of ploughing the circular field)

Total cost of ploughing = $3850 \times 0.5 $= Rs. 1925.

Therefore, Option C is correct.

Note - In these types of problems, simply a common parameter(which is radius here) is calculated from the given data which will help to relate between the two processes which are fencing(circumference based) and ploughing(area based).