Help with fencing question... ASAP

Sep 2019
2
0
Canada
This question from my assignment is giving me some trouble. I feel like I'm really close but my end results still end up screwed up. If anyone can help me solve this so I can hand it in tomorrow afternoon I'd definitely appreciate it :)!

A farmer has 120m of fencing and plans to use it to build a rectangular garden. One side of the garden will be against his house and does not require fencing. What dimensions will allow the maximum area for this garden? What is this maximum area?
 
Oct 2018
129
96
USA
$L$ is the length of the side of the fence opposite the wall of the house, $W$ is the length of the other sides.

$L + 2W=120$
$A=LW$

We can solve for $L$

$L=120-2W$

So,

$A=(120-2W)W$
$A=120W-2W^2$

Now we can use the vertex formula $x_v = \frac{-b}{2a}$ to find the value of $W$ where $A$ is at max.

$\frac{-b}{2a} = \frac{-120}{-4} = 30 = W$

So, we know $W=30$. We use the equation for $L$, $L=120-2W$ to find

$L=120-2(30) = 60$

So the area is at maximum when $L=60$ and $W=30$

Apply the Area formula to find the max area is

$A=60 \times 30 = 1800$
 
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skipjack

Forum Staff
Dec 2006
21,481
2,470
If u and v are real and u + v is a constant, uv = ((u + v)² - (u - v)²)/4 is maximized when u = v.

For the fencing problem, let u = 120 - 2W and v = 2W, then u = v implies 120 - 2W = 2W, so W = 30, etc.
 
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