Question

Fencing a Farmer’s Field As a farmer, suppose you want to fence off a rectangular field that borders a river. You wish to find out the dimensions of the field that occupies the largest area if you have 2000 feet of fencing. Remember that a square maximizes area, so use a square in your work.

- Draw several diagrams to express the situation and calculate the area for each configuration, then estimate the dimension of the largest possible field.

- Find the function that models the area in terms of one of its sides.

- Find the point that maximizes the function you found in the second bulleted item.

- Calculate the area of the field at the point you found in the third bulleted item, and then compare your results with the results in the first bulleted item.

Answer 1

Let x ft. be the length ( measurement of the side along the river) and let y ft. be the width of the rectangular field.

Since the side along the river does not require any fencing, hence the length of fencing required is x+2y . Further, since 2000 feet of fencing is available, hence x+2y = 20000 or, x = (20000-2y).

Now, the area of the rectangular field is A (say) = length *width = xy = (20000-2y)y = -2y²+20000y so that dA/dy = -4y+20000 and d²A/dy² =-4 . Now, A will be maximum when dA/dy = 0 and d²A/dy² is negative. Here, if dA/dy = 0, then -4y+20000 = 0 or, 4y = 20000 so that y = 20000/4 = 5000. Also, d²A/dy² = -4 is negative regardless of the value of y. Hence the area of the rectangular field will be maximum when y = 5000. Then x = 20000-2y = 20000-10000 = 10000.Then A = xy = 5000*10000 = 50000000.

Thus, with the constraint of 20000 feet of fencing, the dimensions of the largest area are 10000 ft. (length along the river) and 5000 ft. width. The largest area is 50000000 sq. ft.