Final answer:
Jill can use the equation -2a^2 + 120a - 1,800 = 0 to solve for the width 'a' and the length of the paddock when the area is 1,800 square feet and she has 120 feet of fencing, with Option B being the correct choice.
Step-by-step explanation:
To find the dimensions of the paddock, Jill needs to solve for the width a and length of the rectangular paddock using the 120 feet of fencing and the specified area of 1,800 square feet. Since one side of the paddock is the side of a barn, the fencing will only be used for the remaining three sides. Let's call the length of the paddock (the side parallel to the barn) x and the width (the two sides perpendicular to the barn) a. The total length of the fencing used is x + 2a, which equals 120 feet.
We know that the area of the rectangle is given by Area = length × width, which translates to 1,800 = x × a. Combine the two equations: x = 120 - 2a and 1,800 = x × a, by substituting x from the first equation into the second, we get 1,800 = (120 - 2a) × a. After expanding, we have 1,800 = 120a - 2a^2, and then rearranging terms gives us the quadratic equation -2a^2 + 120a - 1,800 = 0. Option B matches this equation.