Question

If 200 feet of fencing is used to enclose a rectangular plot of land that borders a river, what is the maximum area that can be enclosed?

Answer 1

i got 20,000 ft^2

Answer 2

I think the question is saying that the fencing is for the whole border of the land, which is rectangular. It cannot be an area of 20,000 because that could be length 200 * width 200. The 200 feet is the whole perimeter.

Answer 3

When you do length*width to solve for area, there are 2 measurements or length and 2 measurements of width, for all the 4 sides of the rectangle. This can be put into an equation as 2x + 2y = 200., where x can be length and y can be width.

With this equation, you need to find possible values for x and y, that make the equation correct (=200). Then, out of those paired values, mulitply them, and see what pair would make the maximum area.

Answer 4

This is an optimization problem, and we need to maximize the area of the rectangle. In general we have this drawing will give you an idea of how to set up the equation, notice we have \(2x+y=200\) and we use the area formula for a rectangle which is \(A=xy\). The idea here is to eliminate the y variable which should be fairly simple as we can solve for y and plug it into our area formula. This will give us the function we want to maximize, so once you have your function we still need to find the derivative so we get our critical values that will give you your interval for maximum area.

Answer 5

Although it's an optimization problem, there's a very clear nature behind maximum area.

SYMMETRY!

Answer 6

:) yeah

Answer 7

@chris215
This problem asks:
Using a ribbon (that for sure has a definite length) , what kinda rectangle can you make that has biggest area?

Answer 8

Think of a rectangle that is the most symmetrical one you know.

Answer 9

in the picture below, two rectangles have similar perimeters


Which one seems more symmetrical?