1. **State the problem:** We need to find the volume of a cylinder with radius $r=5$ cm and height $h=10$ cm. 2. **Formula for the volume of a cylinder:** $$V=\pi r^2 h$$ where $V$ is the volume, $r$ is the radius, and $h$ is the height. 3. **Substitute the given values:** $$V=3.14 \times 5^2 \times 10$$ 4. **Calculate the square of the radius:** $$5^2=25$$ 5. **Multiply the values:** $$V=3.14 \times 25 \times 10$$ 6. **Simplify step-by-step:** $$V=3.14 \times \cancel{25} \times \cancel{10}$$ $$V=3.14 \times 250$$ 7. **Final calculation:** $$V=785$$ 8. **Round the answer:** The volume is $785.0$ cubic centimeters. **Answer:** The volume of the cylinder is $785.0$ cm$^3$.