a. A rectangular pen is built with one side against a barn. If 1000 m of fencing are used for the other three sides of the pen, what dimensions maximize the area of the pen? b. A rancher plans to make four identical and adjacent rectangular pens against a barn, each with an area of 25 m² (see figure). What are the dimensions of each pen that minimize the amount of fence that must be used? Barn 25 25 25 25 a. Let A be the area of the rectangular pen and let x be the length of the sides perpendicular to the barn. Write the objective function in a form that does not include the length of the side parallel to the barn. A = x(1000-2x) (Type an expression.) The interval of interest of the objective function is [0,500] . (Simplify your answer. Type your answer in interval notation. Do not use commas in the individual endpoints.) To maximize the area of the pen, the sides perpendicular to the barn should be 250 m long and the side parallel to the barn should be 500 m long. (Type exact answers, using radicals as needed.) b. Let x be the length of the sides perpendicular to the barn and let L be the total length of fence needed. Write the objective function. 100 L=5x+ X (Type an expression.) The interval of interest of the objective function is (0,00). (Simplify your answer. Type your answer in interval notation. Do not use commas in the individual endpoints.) To minimize the amount of fence that must be used, each of the sides perpendicular to the barn should be ☐ m long and each of the sides parallel to the barn should be m long. (Type exact answers, using radicals as needed.)