optimizing quadratic anonymous functions word problem

Question

Rancher has 1000 feet of fencing he wants to create a rectangular garden area with one side bounded by the river and the other three sides bounded by fencing he also wants to insert an inner piece of fence to divide the rectangular region into two seperate fields. let x represent the lengths of the fences perpendicular to the river and let y represent the length of the fence parallell to the river.

A. let a represent the area of the region the farmer wishes to maximize. state a formula for a in terms of x and y.

B. State a formula that expresses the total amount of fencing used in terms of x and y.

C. Use the result from part B to eliminate either the x or y from the formula in partA

Answer 1

A) a=xy is the total rectangular area of the two fields. The area of each field is ½xy. If the area of each field is maximised it will be square of side x, so the total area of the fields is a=2x².

B) 3x+y=1000 is total amount of fencing split between the parallel and perpendicular fences.

C) If the total area is maximised, it will be a square so x=y and 3x+x=1000, so x=y=1000/4=250 feet.

If the area of each field is maximised, y=2x so that each field is square, and 3x+2x=1000, so x=1000/5=200 feet. The perpendicular fences are 200 feet long are the parallel fence is 400 feet long. The combined area will be 80,000 square feet, in a rectangle which is 400 by 200 feet.