### Problem Description#### a. A rectangular pen is built with one side against a barn. If 300 m of fencing is 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 100 m² (see figure). What are the dimensions of each pen that minimize the amount of fence that must be used?### DiagramsThe diagram shows a barn with four adjacent rectangular pens. Each pen has an area of 100 m². The pens are placed in a row, sharing sides with each other and against the barn.### Objective Function#### 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.\[ A = \, \](Type an expression.)

a) Let x represents the width of the pen and y represents the length of the pen. Given that, the length of the fence for the other three sides is 300 m. Therefore, y=300-2x. Obtain the area of the pen A=l×w. A=x×300-2x=300x-2x2Differentiate the A=300x-2x2 with respect to x as follows, dAdx=ddx300x-2x2=300-4x Equate the derivative to 0. 300-4x=04x=300x=3004x=75 Thus, the width of the pen is 75 m and the length of the pen is y=300-2x. y=300-275=150 Therefore, the dimension of the pen is 150 m by 75 m.b) Let x represents the length of the pen and y represents the width of the pen. Therefore, the total length of the pen is L=5x+4y Given that, the area of each pen is 100 and each are identical and adjacent against a barn. That is, Area A=x×y. x×y=100y=100x Substitute y=100x in L=5x+4y. L=5x+4100x=5x+400x Differentiate the L=5x+400x with respect to x. dLdx=ddx5x+400x=5-400x2 Equate the derivative to zero to find the critical point. 5-400x2=0400x2=55x2=400x2=80x=±45 Note that, the length is always positive, so ignore the root x=-45. Thus, the length of the each pen is 45. The width of the pen is y=40045. y=10045=255=55 Therefore, the dimension of the each pen is 45 by 55.