A farmer has 1600 ft of fencing and wants to fence off a rectangular field that borders a straight river. He does not need a fence along the river

Question

Consider the following problem: A farmer has 1600 ft of fencing and wants to fence off a rectangular field that borders a straight river. He does not need a fence along the river (see the figure). What are the dimensions of the field of largest area that he can fence?

(a) Experiment with the problem by drawing several diagrams illustrating the situation. Calculate the area of each configuration, and use your results to estimate the dimensions of the largest possible field.

(b) Find a function A that models the area of a field in terms of one of its sides x.

A(x) =

(c) Use a graphing calculator to find the dimensions of the field of largest area. Compare with your answer to part (a).

width     ft

length     ft

Answer 1

You need to set up a perimeter equation first, solve for the length (L), then you can answer the first question.

1600 = (2)x + L, now solve for L

L = 1600 - 2x

For Area, it would just be x *L and L is noted above, so Area = x(1600 - 2x) = 1600x -2x²

Answer 2

Let L= length

Let W= width = (1600-L)/2

A=LW = L(1600-L)/2= 800L-L^2/2

take the first derivative and set = 0

A'=800- L = 0

L=800

2W=1600-800= 800

W=400

Largest area is 800 x 400 = 320,000