1. **State the problem:** We are given a circle with center D, radius 14 inches, and a sector formed by points E and F with central angle $\angle EDF = 48^\circ$. We need to find the area of the sector. 2. **Formula for the area of a sector:** The area $A$ of a sector with radius $r$ and central angle $\theta$ (in degrees) is given by: $$ A = \frac{\theta}{360} \times \pi r^2 $$ 3. **Substitute the known values:** Here, $\theta = 48^\circ$ and $r = 14$ inches. $$ A = \frac{48}{360} \times \pi \times 14^2 $$ 4. **Simplify the fraction:** $$ \frac{48}{360} = \frac{48 \div 12}{360 \div 12} = \frac{4}{30} = \frac{2}{15} $$ 5. **Calculate the radius squared:** $$ 14^2 = 196 $$ 6. **Calculate the area:** $$ A = \frac{2}{15} \times \pi \times 196 = \frac{2 \times 196}{15} \pi = \frac{392}{15} \pi $$ 7. **Final answer:** The area of the sector is: $$ \boxed{\frac{392}{15} \pi \text{ square inches}} $$ This matches the option $\frac{13}{15} (196\pi)$ if we simplify or check equivalence, but our exact calculation is $\frac{392}{15} \pi$. Note: $\frac{392}{15} = 26.13$ approximately, and $\frac{13}{15} \times 196 = \frac{13 \times 196}{15} = \frac{2548}{15} = 169.87$ which is different, so the correct area is $\frac{392}{15} \pi$. Hence, the correct area is $\frac{392}{15} \pi$ square inches.