### Solution:

Given, the increment in the area of a circle is from $\n \n 9\pi \n $ to $\n \n 16\pi \n $ .We have to find the ratio of the circumference of the first circle to the second circle.

We know that area of the circle is $\n \n A=\pi \n r\n 2\n \n \n $ .

Let for circle 1, the area of the circle is $\n \n \n A\n 1\n \n =\pi \n r\n 1\n \n \n \n 2\n \n \n $ and for circle 2, the area of the circle is $\n \n \n A\n 2\n \n =\pi \n r\n 2\n \n \n \n 2\n \n \n $ .

We get,

$\n \n \n \n \n A\n 1\n \n =\pi \n r\n 1\n \n \n \n 2\n \n \n \n \n \n \n \u21d29\pi =\pi \n r\n 1\n \n \n \n 2\n \n \n \n \n \n \n \u21d29=\n r\n 1\n \n \n \n 2\n \n \n \n \n \n \n \u21d2\n r\n 1\n \n =3\n \n \n \n \n $

Also,

$\n \n \n \n \n A\n 2\n \n =\pi \n r\n 2\n \n \n \n 2\n \n \n \n \n \n \n \u21d216\pi =\pi \n r\n 2\n \n \n \n 2\n \n \n \n \n \n \n \u21d216=\n r\n 2\n \n \n \n 2\n \n \n \n \n \n \n \u21d2\n r\n 2\n \n =4\n \n \n \n \n $

Now, the circumference of a circle is given by $\n \n C=2\pi r\n $ .

Circumference of circle 1 is $\n \n \n C\n 1\n \n =2\pi \n r\n 1\n \n \n $ .

$\n \n \n \n \n C\n 1\n \n =2\pi \n r\n 1\n \n \n \n \n \n \n \u21d2\n C\n 1\n \n =2\pi \xd73\n \n \n \n \n \u21d2\n C\n 1\n \n =6\pi \n \n \n \n \n $

Also,

$\n \n \n \n \n C\n 2\n \n =2\pi \n r\n 2\n \n \n \n \n \n \n \u21d2\n C\n 2\n \n =2\pi \xd74\n \n \n \n \n \u21d2\n C\n 2\n \n =8\pi \n \n \n \n \n $

Now, the ratio of the circumference of the first circle to the second circle is given by,

$\n \n \n \n \n C\n 1\n \n :\n C\n 2\n \n \n \n \n \n \n \u21d26\pi :8\pi \n \n \n \n \n \u21d23:4\n \n \n \n \n $

Therefore, the ratio of the circumference of the first circle to the second circle is 3: 4.

Hence, the correct option is 1.