Question

The cost of fencing a circular field at the rate of Rs 24 per metre is Rs 5820. Find
[i] The radius of the circle
[ii] The cost of ploughing the field at the rate of Rs. 2 per square metre.

Answer 1

Hint: Assume that the radius of the circle is r. Find the circumference of the circle using the formula for the circumference of the circle and hence find the cost of fencing the field in terms of r. Equate this expression to the given cost and hence form a linear equation in r. Solve for r. The value of r will be the radius of the circle. Now find the area of the circle using the formula for calculating the area of the circle and hence find the cost of ploughing the field at the given rate.

Complete step-by-step answer:
Let r be the radius of the circle.
So, we have the circumference of the circle is $2\pi r$ because the circumference of a circle with radius r is given by $2\pi r$.
Cost of fencing 1 metre = Rs. 24
Hence the cost of fencing $2\pi r$ metres $=24\times 2\pi r=48\pi r$
But given that the cost of fencing the field is Rs. 5280.
Hence, we have
$48\pi r=5280$
Dividing both sides by $48\pi $, we get
$r=\dfrac{5280}{48\pi }=\dfrac{485}{4\pi }=35$ metres.
Hence the radius of the field is 35 metres.
Now, we know that area of the circle with radius r is given by $\pi {{r}^{2}}$
Hence the are of the circle of radius 35 is given by
$\pi {{\left( 35 \right)}^{2}}=3850$ square metres.
Cost of ploughing 1 square metre = Rs. 2
Hence, the cost of ploughing 3850 square metres $=2\times 3850=7700$
Hence the cost of ploughing the field = Rs. 7700

Note: [1] In the solution, we have used an approximated value of $\pi =\dfrac{22}{7}$. Unless stated otherwise $\pi $ is always taken as $\dfrac{22}{7}$.
[2] Students often get confused over whether to find the area or the perimeter. It can be understood from the dimensions of the rates also, whether to find the area or the perimeter. If the rate is per square unit, then the quantity is dependent on area, and if the rate is per unit, then the quantity is dependent on length(perimeter).