The problem states that the length of an arc of a circle of radius $3.5$ cm is $1 \frac{19}{36}$ cm. We need to calculate the angle subtended by the arc at the center of the circle, correct to the nearest degree, using $\pi = \frac{22}{7}$.

1. Problem Description

The problem states that the length of an arc of a circle of radius $3.5$ cm is $1 \frac{19}{36}$ cm. We need to calculate the angle subtended by the arc at the center of the circle, correct to the nearest degree, using $\pi = \frac{22}{7}$.

2. Solution Steps

First, convert the mixed fraction to an improper fraction:

$1 \frac{19}{36} = \frac{1 \times 36 + 19}{36} = \frac{36 + 19}{36} = \frac{55}{36}$ cm.

The formula relating the length of an arc ($l$), the radius ($r$) and the angle subtended at the center ($\theta$ in radians) is given by:

$l = r\theta$

Therefore, $\theta = \frac{l}{r}$.

Substituting the given values, we get:

$\theta = \frac{\frac{55}{36}}{3.5} = \frac{55}{36 \times 3.5} = \frac{55}{36 \times \frac{7}{2}} = \frac{55}{18 \times 7} = \frac{55}{126}$ radians

To convert radians to degrees, we use the formula:

Degrees = Radians $\times \frac{180}{\pi}$

Degrees $= \frac{55}{126} \times \frac{180}{\frac{22}{7}} = \frac{55}{126} \times \frac{180 \times 7}{22} = \frac{55 \times 180 \times 7}{126 \times 22} = \frac{5 \times 180 \times 7}{126 \times 2} = \frac{5 \times 180 \times 1}{18 \times 2} = \frac{5 \times 10 \times 1}{2} = 5 \times 5 = 25$ degrees.

3. Final Answer

The angle subtended by the arc at the center of the circle is $25$ degrees.

So, the correct answer is C.