Question

Consider the following problem: A breeder of horses wants to fence three rectangular grazing fields along the river with 1800 total feet of fencing as shown below. (Let x denote the vertical length of each of two sides and two dividers. Let y denote the horizontal length for each of the three grazing areas.) What is the largest possible total area (for all three grazing fields together) that she can enclose? What would the dimensions be for each grazing field? Please make sure to VERIFY that you have found a maximum or minimum value with the first or second derivative test! (a) Write an expression for the area of one grazing field A in terms of both x and y. (b) Use the given information to write an equation in terms of only x and y that relates the variables. (c) Use part (b) to write the area of one grazing field as a function of one variable. A(x) = (d) Finish solving the problem by finding the dimensions for each grazing field (x and y). Then find the largest possible total area for all three grazing fields put together. x = ft y = Total Area for all three grazing fields = ft2

Answer 1

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@ A= my 6 Total length of fenu = 4x+3y - - 14x+3y = 1800 ft © A= xy = x [ 1800 - 4x = 600 x - 4 x ² A(x) = 600 4-4 u & For max. (3A) [total area = A +A+A 23A ] der a=3A, da so do so do = 3 2 A 3[ 600 - 1, 2x] = 3 [ 600 x da = o > 600 = bx au boux3 = 225 y = 180-4x _ 600-4 ( 25) = 300 - on 2 = 3[0-93] = -8 <0. 75 X = 225 17 Y= 300 ft Total area of grazen fieles- 3 xy = 3 (225 x 310) fj2 = 202500 f2