1. **State the problem:** Calculate the volume of a cone with base radius $r=5$ cm and curved surface area $A=109.9$ cm$^2$. 2. **Recall formulas:** - Curved surface area of a cone: $$A = \pi r l$$ where $l$ is the slant height. - Volume of a cone: $$V = \frac{1}{3} \pi r^2 h$$ where $h$ is the height. 3. **Find the slant height $l$:** $$l = \frac{A}{\pi r} = \frac{109.9}{\pi \times 5} = \frac{109.9}{15.707} \approx 7$$ cm 4. **Find the height $h$ using Pythagoras theorem:** $$h = \sqrt{l^2 - r^2} = \sqrt{7^2 - 5^2} = \sqrt{49 - 25} = \sqrt{24} \approx 4.899$$ cm 5. **Calculate the volume $V$:** $$V = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi \times 5^2 \times 4.899 = \frac{1}{3} \pi \times 25 \times 4.899 \approx \frac{1}{3} \times 3.1416 \times 122.475 = 128.7$$ cm$^3$ **Final answer:** The volume of the cone is approximately $128.7$ cubic centimeters.