Video solution:

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Written solution:

Since $FE$ is the diameter of the semicircle, we know the length of the diameter is $34,$ and so the radius is $17.$ Let $O$ be the center of the diameter.

The length from $OF$ therefore is $17.$

Since $D$ is on $OF,$ we know $$\begin{align*} OD + FD &= OF\\ OD + 9 &= 17 \\ OD &= 8. \end{align*}$$

Also, since we have a semicircle, we know $OC = 17.$

Finally, since $ABCD$ is a rectangle, we know $\angle ODC$ is a right angle. This means we can find $DC$ by the Pythagorean Theorem. We know $$\begin{align*} OD^2+DC^2&=OC^2 \\ 8^2+DC^2 &= 17^2 \\ DC &= 15. \end{align*}$$

Thus, the area of the rectangle is $DC\cdot DA = 15\cdot 16=240.$

Thus, the correct answer is A.