Optimizing Fencing for a Rectangular Enclosure with a Fixed Wall

[lionely]

Homework Statement

A man wishes to fence in a rectangular enclosure of area 128m^2.One side of the enclosure is formed by part of a brick wall already in position.
What is the least possible length of fencing required for the other three sides?

2. Homework Equations

The Attempt at a Solution

I called one side x and one side y.

So far I have the relationship

xy = 128
y = 128/x

I was going to differentiate that but that won't help me since equating the result to 0 give me nothing.

I was thinking the perimeter might help me but.. I don't know what to do with it. Help is greatly appreciated

 

[Orodruin]

I was thinking the perimeter might help me but.. I don't know what to do with it. Help is greatly appreciated

Your problem is to minimise the length of the fence. A good place to start is to find an expression for the length of the fence.

 

[lionely]

Okay i'll call the length L , so L = 2x + y?

 

[Orodruin]

Correct so far. Now what is the minimal L which encloses 128 m²?

 

[lionely]

okay so I did this L = 2x + y .
L = 2x + (128/x) [from my first area expression]
L = (2x^2 + 128)/x
dL/dx = -(2x^2 + 128)/x^2 + 4

For max/min dL/dx = 0
2x^2 + 128 = 4x^2
x = +/- 8
For L to be a minimum x= 8.
So Lmin = 16 + 16 = 32m

 

[Orodruin]

For L to be a minimum x= 8.

Yes, there is also no way you can make a side with length -8 m. Be mindful of the domains you have to consider.

It is also easier to just differentiate 2x + 128/x directly than rewriting it as (2x^2 + 128)/x. The result will of course be the same.

 

[lionely]

Oh yeah true, haha wasn't thinking. I truly appreciate it!