Question

The areas of two circular fields are in the ratio 16:49. If the radius of the latter is 14 m, then what is the radius of the former?
(a) 10 m
(b) 8 m
(c) 12 m
(d) 16 m

Answer 1

Hint: By looking at the question we directly know that we have to use the area of the circle which is \[\pi {{r}^{2}}\], where \[r\] is the radius of the circle. Here we have to divide the area of two circular fields.

Complete step-by-step Solution:
Before proceeding with the question, we should know the definition of circle. A circle is a round plane figure and it forms a closed loop. It has a center and all the points are equidistant from it. Area of a Circle is: \[A=\pi {{r}^{2}}.............(1)\]
As the radius of the second circle is given we assume the radius of the first circle to be \[r\].
Using equation (1) we find the areas of two circles.
So area of first circle is: \[{{A}_{1}}=\pi \times {{r}^{2}}..........(2)\]
As radius of second circle is given and it is 14 m, so
area of second circle is: \[{{A}_{2}}=\pi \times {{14}^{2}}.............(3)\]
And now it is given in the question that areas of two circles are in the ratio 16:49, so dividing equation (2) with equation (3) we get,
\[\dfrac{{{A}_{1}}}{{{A}_{2}}}=\dfrac{\pi \times {{r}^{2}}}{\pi \times {{14}^{2}}}..........(4)\]
Now equating equation (4) to the ratio given in the question we get,
\[\Rightarrow \dfrac{\pi \times {{r}^{2}}}{\pi \times {{14}^{2}}}=\dfrac{16}{49}\]
Now cancelling the similar terms we get,
\[\Rightarrow \dfrac{{{r}^{2}}}{{{14}^{2}}}=\dfrac{16}{49}\]
Now taking square root of both sides we get,
\[\Rightarrow \dfrac{r}{14}=\dfrac{4}{7}\]
Now taking all the numbers to one side,
\[\Rightarrow r=\dfrac{4}{7}\times 14=4\times 2=8\,\text{m}\].
Hence 8 m is the answer. So option (b) is the right option.

Note: In a hurry we can make a mistake by taking 14 m as radius of the first circle and hence reading the question two to three times is important. We also need to be careful with units as here it is in meters but sometimes they can give one radius in meters and another in centimeters. At the time of taking square root we need to consider only positive values.