Question

The outer circumference of a circular racing track is 220 m. The track is everywhere 7 m wide the cost of levelling the track at the rate of 50 paise per square meter (Use $\pi = \dfrac{{22}}{7}$ ) is:
A) Rs 540
B) Rs 650
C) Rs 693
D) Rs 700

Answer 1

Hint:
Use the formula for finding the area of the circular track, given by $A = \pi \left( {{R^2} - {r^2}} \right)$ , when the outer radius ‘R’ and inner radius ‘r’ is involved. After finding the area of the required track multiply it with the rate at which the track is levelled by changing the units by the conversion methods, wherever necessary. Also, construct the figure using the given information.

Complete step by step solution:
we will construct the figure first,

Let us begin with denoting the outer circumference by R and the inner radius by r.
It is known that the circumference of a circle is given by the formula, $C = 2\pi r$. Given that the outer circumference is 220 m.
$
   \Rightarrow 220 = 2\pi R \\
   \Rightarrow R = \dfrac{{220}}{{2\pi }} \\
   \Rightarrow R = \dfrac{{220 \times 7}}{{2 \times 22}} \\
   \Rightarrow R = 35{\text{ m}} \\
 $
Now according to the question we see that the width of the track is 7 m, thus we find the radius of the inner circle, by subtracting 7m from the outer radius.
$ \Rightarrow 35{\text{ m}} - 7{\text{ m}} = 28{\text{ m}}$
We will now move on to calculating the area of our circular track using the formula, $A = \pi \left( {{R^2} - {r^2}} \right)$.
We substitute $\pi = \dfrac{{22}}{7}$, $R = 35$ and $r = 28$ .
$
   \Rightarrow A = \dfrac{{22}}{7}\left( {{{35}^2} - {{28}^2}} \right) \\
   \Rightarrow A = \dfrac{{22}}{7}\left( {35 - 28} \right)\left( {35 + 28} \right) \\
   \Rightarrow A = \dfrac{{22}}{7} \times 7 \times 63 \\
   \Rightarrow A = 1386{\text{ }}{{\text{m}}^2} \\
 $
Now, going back to the question, we note that we need to find the cost of levelling the track where the rate of levelling the track is 50 paise per square meter or we can say it is Rs 0.50 per square meter.
Thus, we get the cost of levelling the track by multiplying the area of the circular track by is Rs 0.50 per square meter.
\[
   \Rightarrow {\text{Cost}} = 1386 \times 0.50 \\
   \Rightarrow {\text{Cost }} = {\text{Rs. }}693 \\
 \]

Hence, the cost of levelling the track is Rs. 693. Hence, option (C) is the correct option.

Note:
Units of the amount given in the options and what you calculated should be the same. Always apply the conversion factors wherever necessary. Always take care of the calculation errors in these types of questions. Also, use the value of $\pi $, which is mentioned in the question, otherwise, you might not get the correct answer.