Question

The inner circumference of a circular track is $440cm$. The track is $14cm$ wide. Find the diameter of the outer circle of the track.

Answer 1

Hint: In this question we have to find the diameter of the circle. So thus we have to first find the radius of the outer circle, because we know that diameter is twice of the radius i.e. $d = 2r$ . We will find the radius using the formula of circumference of the inner circle. The formula of circumference is $2\pi r$ .

Complete step by step solution:
In this question we have given the width of the track, which means that it is the difference between the inner radius to the outer radius of the circle.
Let us assume the radius of inner circle be $r$
And the radius of the outer circle is $R$ .
We will now draw the diagram representing these data:

Here in the above figure OA is the radius of the inner circle i.e.
 $OA = r$
And the radius of outer circle is
$OB = R$ .
It is given in the question that The track is $14cm$ wide .
It can also be written as
 $R - r = 14\,cm$
Now we can calculate the circumference of the inner circle by the formula $2\pi r$ ,
But we have been given the circumference of the inner circle i.e. $440cm$ .
So by substituting this back in the formula, we can write
$440 = 2 \times \dfrac{{22}}{7} \times r$
We will now simplify this:
$440 = \dfrac{{44r}}{7}$
On cross multiplying the values, we have
$r = \dfrac{{440 \times 7}}{{44}}$
Therefore it gives the value
$r = 70cm$ .
We will put this value in the expression of track width i.e.
$R - 70 = 14$
It gives us the value of outer radius i.e.
$R = 70 + 14 = 84cm$
Now we know that diameter is twice the radius, so we can calculate the diameter i.e.
$D = 2 \times 84$
It gives us value $168cm$
Hence the required answer is $168cm$ .
So, the correct answer is “$168cm$”.

Note: We should be careful that while subtracting the inner radius to outer radius, we should not directly subtract the circumferences of two circles because it is wrong. Also if we have been asked to find the area of the track, then we have to get both the radius of the circle and use the formula: $Area = \pi \left( {{R^2} - {r^2}} \right)$ .