1. **State the problem:** We are given the area of a circle as $\frac{9}{16}$ square inches and need to find the radius of the circle, rounded to the nearest thousandth inch. 2. **Formula used:** The area $A$ of a circle is related to its radius $r$ by the formula: $$A = \pi r^2$$ 3. **Substitute the given area:** $$\frac{9}{16} = \pi r^2$$ 4. **Solve for $r^2$:** $$r^2 = \frac{9}{16\pi}$$ 5. **Take the square root to find $r$:** $$r = \sqrt{\frac{9}{16\pi}} = \frac{3}{4\sqrt{\pi}}$$ 6. **Simplify and approximate:** $$r \approx \frac{3}{4 \times 1.77245385} = \frac{3}{7.0898154} \approx 0.423$$ 7. **Final answer:** The radius needed is approximately **0.423 inch** to the nearest thousandth.