Can you show me the steps to get these answers for my practice? a. A rectangular pen is built with one side against a barn. If 2500 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 m (see figure). What are theВamdimensions of each pen that minimize the amount of fence that must be used?2525a. 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 sideparallel to the barn.A = 2500x – 2x2(Type an expression.)The interval of interest of the objective function is [0,1250] .(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 625 m long and the side parallel to the barn should be 1250 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.100L= 5x +(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.)5/20To minimize the amount of fence that must be used, each of the sides perpendicular to the barn should be v20 m long and each of the sides parallel to the barn should be4long.(Type exact answers, using radicals as needed.)

Note: According to Bartleby guidelines; For more than one question asked only the first one is to be answered. Given data: Length of the barn requires to fence the three sides of the pen=2500 m Let x be the measure of the side perpendicular to the barn and y be the measure of the side parallel to the barn. Let y be the measure of the side against the barn. Answer for sub question 1: Use the data that the length of the barn required to fence the three sides=2500m, to write, 2x+y=2500→1 The objective of the given question is to maximize the area. Recall: The formula to find the area of a rectangle with length x and breadth y is A=xy Therefore, the area of the rectangular pen is, A=xy But, it has been asked to express the objective functionArea function in a form that does not include the length of the side parallel to the barn. Therefore, the objective function must not involve y (the length of the side parallel to the barn). Hence express y in terms of x by using equation 1. From 1, 2x+y=25002500-2x=y→2 Now substitute for y in the Area function. A=xy=x2500-2x=2500x-2x2 Observe that now the area function does not involve y(the length of the side parallel to the barn) Hence, A=2500x-2x2 is the answer for sub question 1. Answer for sub question 2: The length of the side perpendicular to the barn cannot be less than zero and hence the minimum value of x(the length of the side perpendicular to the barn) must be zero. Also the length of the side parallel to the barn cannot be less than zero and hence the minimum value of y(the length of the side perpendicular to the barn) must be zero. When y takes the minimum value, x takes the maximum value. Substitute y=0 in 2, to get the maximum value of x. 2500-2x=y2500-2x=02500=2x25002=x1250=x Therefore, the interval of interest of the objective function is 0,1250 Answer for sub question 3: Consider the objective function: A=2500x-2x2 The maximum area occurs when dAdx=0 First find dAdx: dAdx=2500-4x Equate dAdx=0: dAdx=02500-4x=02500=4x25004=xx=625 Therefore the maximum area occurs when x=625. Therefore the length of the side perpendicular to the barn is 625m. To find the length of the side parallel to the barn substitute x=625 in 2. y=2500-2x=2500-2625=2500-1250=1250 Therefore the length of the side perpendicular to the barn is 1250m.