Question

Warning: Really confusing and hard word problem

Enclosing the most area with a fence: A farmer with 10,000 meters of fencing wants to enclose a rectangular field and then divide it into two plots with a fence parallel to one of the sides. What is the largest area that can be enclosed?

Answer 1

let the length = x

and assume the 3rd parallel fence is also x

then the fencing is

10000 = 3x + 2w

so then make w the subject of the equation

2w = 1000 - 3x

or \[w = \frac{10000 - 3x}{2}~~~or~~~~w = 5000 - \frac{3}{2} x\]

so now you have the dimensions

length = x

width = 5000 - 3/2 x

so the area is

\[A = x(5000 - \frac{3}{2} x)\]

now distribute to find the area in terms of x.

So if this is a calculus question,

1. find the derivative

2. solve the derivative for x

3. this will be the length that gives the maximum area.

4. find the width from the equation above...

5. find the area.

hope it makes sense.

Answer 2

here is the diagram

Answer 3

The idea is that the slope of a function at a max or min is zero.

So if your function represents the total area, taking its derivative and setting it to 0 will give you the maximum (or minimum) area.

Answer 4

Thanks guys :)

Answer 5

By the way, I'm only in pre-calc so I haven't learned about derivatives yet. I can just graph the function and find the min/max. But derivatives sounds like a really cool concept!

Answer 6

if it's not a calculate question just graph the area function... and then the highest point will be the max area and the x value that gives the max area is the length

Answer 7

the only problem with graphing will be the dimensions of the grid....

you will be graphing area as the vertical length as the horizontal...

so good luck with that.... so the vertical axis will need to go to about 5000000

and the horizontal up to 10000 at a guess

Answer 8

Your equation is a parabola, so its vertex is the point you want.