Can you check my work for this math problem?

Question: A rectangular plot of land on the edge of a river is to be enclosed with fence on three sides. Find the dimensions of the rectangular enclosure of the greatest area if the side that goes along the river doesnt require fencing and the total length of the fence is 200 m.





Answer: let x be the side along the river and y be the perpendicular side. Then x +2y=200


whence


x=200-2y


The area of the rectangle is


xy=(200-2y) y= -2y^2 +200y


The greatest value of this quadratic function will be at the vertex


y= -200 / 2 times (-2) = 50


then x=200-2 times 50 = 100





did i get that right?


if not,where did i mess up?Can you check my work for this math problem?
Hi,





You are correct. It's 100 long and 50 feet to the river bank.





One guy incorrectly told you that a square will always have the maximum area for a quadrilateral. That is true if you are fencing all 4 sides, but it is not true here.





There is a neat fact that makes this problem really easy though. If you think of the square, where half the perimeter is used up by the lengths and half is used up by the widths, the same thing will still happen in this rectangle where you are only fencing 3 sides. Half of the fence or 100 feet is the length of the side parallel the river while the other half of the fence is split in 2 for the 2 sides going to the riverbank. Therefore 200/2 = 100 is the length and 100/2 = 50 is each width. This produces the maximum area.





Suppose instead you were making 4 equal sized dog pens up against the side of a kennel so you have to fence one side parallel to the kennel and make 5 walls total that are perpendicular to the kennel to split the pens and enclose the dogs. If you have 300 feet of fence you would use half of it or 150 feet for the side parallel the kennel. the other 150 feet is divided into 5 walls of 30 feet each out away from the kennel. This means the overall area is 150 x 30 or 4500 square feet. Suppose you made the length 160 feet instead. then the other 140 feet divided for 5 walls is 28 feet each. Total area would be 160 x 28 = 4480 square feet, a smaller total area. Suppose you made the length only 140 feet instead. then the other 160 feet divided for 5 walls is 32 feet each. Total area would be 140 x 32 = 4480 square feet, a smaller total area. So half for the total length and half the fence for all the widths will always produce the maximum area.








I hope that helps!!! :-)Can you check my work for this math problem?
I worked this through and got y = 50 and x=100 too, and your reasoning is right too. So it looks right to me.
If I am reading the question correctly,,then sorry,,You are wrong..the question states ';Find the dimensions of the rectangular enclosure of the greatest area'; A square has the greatest area for a given perimeter., Therefore the greatest area you could fence in,,keeping the numbers relatively even, is a fence layout that is 66.5ft. x 67ft x 66.5ft. = 200ft. You can get even closer by working in inches,,to get the very max,, Hope this helps