1. **State the problem:** We need to find the lateral surface area of a square pyramid with base side length $3.11$ m and height $7.22$ m. 2. **Formula for lateral surface area of a square pyramid:** The lateral surface area $A_{lat}$ is given by $$A_{lat} = \frac{1}{2} P l$$ where $P$ is the perimeter of the base and $l$ is the slant height. 3. **Calculate the perimeter $P$ of the base:** Since the base is a square with side $3.11$ m, $$P = 4 \times 3.11 = 12.44 \text{ m}$$ 4. **Find the slant height $l$:** The slant height is the hypotenuse of a right triangle with height $7.22$ m and half the base side as the other leg: $$l = \sqrt{7.22^2 + \left(\frac{3.11}{2}\right)^2} = \sqrt{7.22^2 + 1.555^2}$$ Calculate inside the square root: $$7.22^2 = 52.1284, \quad 1.555^2 = 2.4190$$ So, $$l = \sqrt{52.1284 + 2.4190} = \sqrt{54.5474} = 7.386 \text{ m (approx)}$$ 5. **Calculate the lateral surface area:** $$A_{lat} = \frac{1}{2} \times 12.44 \times 7.386 = 6.22 \times 7.386 = 45.93 \text{ square meters}$$ **Final answer:** The lateral surface area of the square pyramid is approximately **45.93 square meters**.