A rancher has 800 feet of fencing to put around a rectangular field and then subdivide the field into 2 identical smaller rectangular plots by placing a fence parallel to one of the field's shorter sides. Find the dimensions that maximize the enclosed area. Write your answers as fractions reduced to lowest terms.

Answer:The dimensions of enclosed area are 200 and 400/3 feetStep-by-step explanation:* Lets explain how to solve the problem- There are 800 feet of fencing- We will but it around a rectangular field - We will divided the field into 2 identical smaller rectangular plots  by placing a fence parallel to one of the field's shorter sides- Assume that the long side of the rectangular field is a and the  shorter side is b∵ The length of the fence is the perimeter of the field∵ We will fence 2 longer sides and 3 shorter sides∴ 2a + 3b = 800- Lets find b in terms of a∵ 2a + 3b = 800 ⇒ subtract 2a from both sides∴ 3b = 800 - 2a ⇒ divide both sides by 3∴ [tex]b=\frac{800}{3}-\frac{2a}{3}[/tex] ⇒ (1)- Lets find the area of the field∵ The area of the rectangle = length × width∴ A = a × b ∴ [tex]A=(a).(\frac{800}{3}-\frac{2a}{3})=\frac{800a}{3}-\frac{2a^{2}}{3}[/tex]- To find the dimensions of maximum area differentiate the area with   respect to a and equate it by 0∴ [tex]\frac{dA}{da}=\frac{800}{3}-\frac{4a}{3}[/tex]∵ [tex]\frac{dA}{da}=0[/tex]∴ [tex]\frac{800}{3}-\frac{4}{3}a=0[/tex] ⇒ Add 4/3 a to both sides∴ [tex]\frac{800}{3}=\frac{4}{3}a[/tex] ⇒ multiply both sides by 3∴ 800 = 4a ⇒ divide both sides by 4∴ 200 = a- Substitute the value of a in equation (1)∴ [tex]b=\frac{800}{3}-\frac{2}{3}(200)=\frac{800}{3}-\frac{400}{3}=\frac{400}{3}[/tex]* The dimensions of enclosed area are 200 and 400/3 feet