Optimization, Maximizing area excluding margins

[Idealistic]

***Note these are not the correct dimensions.


Basically we have a square or rectangle within a square or rectangle...

The total area of a square/rectangle has an area of 2400 cm^2. If margins 4 cm on each side(left side, right side) and margins 6 cm (top, bottom) are created what is the maximum area of a square/rectangle capable of being produced within those margins; give its dimensions.

The margins are constant, and the total area is, trying to find maximum area of box within box.

So the equations to work with are:

2400 = (x + 8)(y + 12); (area of total square/rectangle) = (total length horizantally x total length vertically)

And:

A = xy; (area of square/rectangle within margins) = (length horizontally minus margins x length vertically minus margins)

Basically what i did was rearranged for x to get, y = 2400/(x + 8) - 12 and pluged it into A = xy. Then I took the derivative but it got me nowhewre fast. Im guessing I have the wrong equations?

Rest of work:

A = x(2400/(x + 4)) -12
A = 2400x/(x + 4) - 12
A' = 2400(x + 4) - 2400x,
A' = 4?

 

[arthur ohlsten]

A=xy

2400=[x=8][y+12]
solve for y
y= 2400/[x+8] - 12
substitute into A=xy
2400 = x [ 2400/[x+8] -12]
2400= 2400x/[x+8] -12x

improper fraction divide [x+8] into 2400x
2400= 2400 -19,200/[x+8] -12x

take derivative with respect to x and set = 0
0= 0+19,200/[x+8]^2 -12
12= 19,200/[x+8]^2
[x+8]^2= 1600
x+8=40
x=32
y=2400/32 = 300/4
y=75

You do it without reducing the improper fraction.
derivative of a fraction = bottom times derivative of top = top times derivative bottom over the bottom squared
And someone will correct me to say numerator and denominator.

Arthur

 

[Idealistic]

A=xy

2400=[x=8][y+12]
solve for y
y= 2400/[x+8] - 12
substitute into A=xy
2400 = x [ 2400/[x+8] -12]
2400= 2400x/[x+8] -12x

improper fraction divide [x+8] into 2400x
2400= 2400 -19,200/[x+8] -12x

take derivative with respect to x and set = 0
0= 0+19,200/[x+8]^2 -12
12= 19,200/[x+8]^2
[x+8]^2= 1600
x+8=40
x=32
y=2400/32 = 300/4
y=75

You do it without reducing the improper fraction.
derivative of a fraction = bottom times derivative of top = top times derivative bottom over the bottom squared
And someone will correct me to say numerator and denominator.

Arthur

But after you find y in terms of x, you cant sub 2400 in for "A" in A = xy; because 2400 is the area of the square including the margins. We want the area of the square excluding the margins which has to be smaller than 2400 cm^2.

 

[Idealistic]

Ohh thanks I finally got it.

x = 32, y = 48.