A circle has a chord that goes from #( 3 pi)/2 # to #(7 pi) / 4 # radians on the circle. If the area of the circle is #196 pi #, what is the length of the chord?

To calculate the radius, use the area:

We can apply the Law of Cosines because the chord, the two radii, and the triangle they form:

To find the length of the chord in the circle, first, we need to determine the radius of the circle using the given area formula for a circle: $A = \pi r^2$. Given that the area of the circle is $196\pi$, we have:

$196\pi = \pi r^2$

Solving for $r$, we get:

$r^2 = 196$

$r = 14$

Now that we know the radius of the circle is 14 units, we can calculate the length of the chord using the formula for the length of a chord in a circle, which is given by:

$L = 2r\sin\left(\frac{\theta}{2}\right)$

where $\theta$ is the angle subtended by the chord at the center of the circle.

Given that the chord spans from $\frac{3\pi}{2}$ to $\frac{7\pi}{4}$ radians, the angle $\theta$ subtended by the chord is:

$\theta = \left( \frac{7\pi}{4} - \frac{3\pi}{2} \right) = \frac{\pi}{4}$

Substituting $r = 14$ and $\theta = \frac{\pi}{4}$ into the formula for the length of the chord, we get:

$L = 2 \times 14 \times \sin\left(\frac{\pi}{8}\right)$

$L = 28 \times \sin\left(\frac{\pi}{8}\right)$

Using the trigonometric value $\sin\left(\frac{\pi}{8}\right) \approx 0.383$, we can calculate:

$L \approx 28 \times 0.383$

$L \approx 10.724$

So, the length of the chord is approximately $10.724$ units.