Alake has a rectangular area roped oll where people can swm under a lifegurd's supervision. The sivmming section has an area of \( x^{2}+3 x-40 \) square fect, with the long side paralled to the shore. a. What are passible dmensions of the roped-olf area? Use factonng. b. Hour much rope is needed for the three sides that are not along the beach? Expian. c. The rope used to mark the swimming area is 238 ft long. What is x when the total length of rope is 238 ft ? a. Passible dmensions of the recedefl area weuld be \( \square \) 1. (7ype express ors usimg as the vanadie (Use a comma to sepavate your answors as neoded)

**Step 1. Find the dimensions by factoring the area.** The swimming area is given by \[ x^2 + 3x - 40. \] We factor this quadratic by finding two numbers that multiply to \(-40\) and add to \(3\). These numbers are \(8\) and \(-5\). Thus, we can write \[ x^2 + 3x - 40 = (x + 8)(x - 5). \] So the dimensions of the rectangular area are \(x+8\) feet and \(x-5\) feet. Since the long side is parallel to the shore, we take \(x+8\) (which is greater than \(x-5\) when \(x>5\)) as the side along the beach. **Step 2. Determine the length of rope needed for the three sides not along the beach.** With the side parallel to the shore (of length \(x+8\)) not needing rope, the rope is required for the remaining three sides: the opposite side (length \(x+8\)) and the two sides perpendicular to the beach (each of length \(x-5\)). Therefore, the total length of rope \(R\) is \[ R = (x+8) + 2(x-5). \] Simplify: \[ R = x + 8 + 2x - 10 = 3x - 2. \] **Step 3. Find \(x\) when the rope used is 238 ft long.** We set the total rope length equal to 238 ft: \[ 3x - 2 = 238. \] Solve for \(x\): \[ 3x = 238 + 2 = 240, \] \[ x = \frac{240}{3} = 80. \] **Final Answers:** a. The possible dimensions of the roped-off area are \(x+8\) and \(x-5\) (use \(x+8, x-5\)). b. The amount of rope needed for the three sides is \(3x-2\) feet. c. For a total rope length of 238 ft, \(x = 80\).