1. **State the problem:** Calculate the volume of a cone with radius $r=7$ mm and slant height $l=13$ mm. 2. **Recall the formula for the volume of a cone:** $$V=\frac{1}{3}\pi r^{2}h$$ where $r$ is the radius and $h$ is the height. 3. **Find the height $h$ using the Pythagorean theorem:** Since the slant height $l$, radius $r$, and height $h$ form a right triangle, $$l^{2} = r^{2} + h^{2}$$ Substitute known values: $$13^{2} = 7^{2} + h^{2}$$ $$169 = 49 + h^{2}$$ $$h^{2} = 169 - 49 = 120$$ $$h = \sqrt{120} = \sqrt{4 \times 30} = 2\sqrt{30} \approx 10.954$$ 4. **Calculate the volume:** $$V = \frac{1}{3} \pi (7)^{2} (10.954) = \frac{1}{3} \pi \times 49 \times 10.954$$ $$V = \frac{1}{3} \times 3.1416 \times 49 \times 10.954$$ $$V = \frac{1}{3} \times 1684.5 = 561.5$$ 5. **Final answer rounded to 1 decimal place:** $$\boxed{561.5 \text{ mm}^3}$$