1. **Problem Statement:** Find the surface area of a cone with radius $r=4$ m, height $h=3$ m, and slant height $l=5$ m. 2. **Formula:** The total surface area $A$ of a cone is given by: $$A = \pi r^2 + \pi r l$$ where $\pi r^2$ is the area of the base and $\pi r l$ is the lateral surface area. 3. **Calculate the base area:** $$\pi r^2 = \pi \times 4^2 = 16\pi$$ 4. **Calculate the lateral surface area:** $$\pi r l = \pi \times 4 \times 5 = 20\pi$$ 5. **Add both areas to get total surface area:** $$A = 16\pi + 20\pi = 36\pi$$ 6. **Answer:** The surface area of the cone is $36\pi$ m$^2$.