1. Problem 17: Find the total distance an ant travels from point (3,4) to (6,10), then from (6,10) to (10,18) in the Cartesian plane. 2. To find the distance between two points $(x_1,y_1)$ and $(x_2,y_2)$, use the distance formula: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ This formula comes from the Pythagorean theorem, where the difference in x-coordinates and y-coordinates form the legs of a right triangle. 3. Calculate the distance from $(3,4)$ to $(6,10)$: $$d_1 = \sqrt{(6 - 3)^2 + (10 - 4)^2} = \sqrt{3^2 + 6^2} = \sqrt{9 + 36} = \sqrt{45} = 3\sqrt{5}$$ 4. Calculate the distance from $(6,10)$ to $(10,18)$: $$d_2 = \sqrt{(10 - 6)^2 + (18 - 10)^2} = \sqrt{4^2 + 8^2} = \sqrt{16 + 64} = \sqrt{80} = 4\sqrt{5}$$ 5. Total distance traveled by the ant is the sum: $$d_{total} = d_1 + d_2 = 3\sqrt{5} + 4\sqrt{5} = 7\sqrt{5}$$ 1. Problem 18: Leslie's backyard is a square with area 98 m². Inside it, a green square with area 8 m² contains a tree. We need to find the exact perimeter of one of the rectangular flowerbeds. 2. Since the backyard is a square, its side length is: $$s = \sqrt{98} = 7\sqrt{2}$$ 3. The green square's side length is: $$g = \sqrt{8} = 2\sqrt{2}$$ 4. The flowerbeds are rectangular sections formed by subtracting the green square from the backyard. Assuming the flowerbeds are formed by the leftover area, the length and width of a flowerbed can be found by subtracting the green square's side from the backyard's side: $$l = s - g = 7\sqrt{2} - 2\sqrt{2} = 5\sqrt{2}$$ 5. Since the flowerbeds are rectangular and the problem implies they are squares or rectangles formed by this difference, the perimeter of one flowerbed is: $$P = 2(l + g) = 2(5\sqrt{2} + 2\sqrt{2}) = 2(7\sqrt{2}) = 14\sqrt{2}$$ Final answers: - Distance ant travels: $7\sqrt{5}$ - Perimeter of one flowerbed: $14\sqrt{2}$