Calculus board optimization question
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Calculus board optimization question

[From: ] [author: ] [Date: 11-12-10] [Hit: ]
soA = Prod=x(25-x) =25x - x^2... Thisis aparabola,openingdownwards sothe derivative willgive amaximum.= 25 - 2x andsettingthisderivative to zero : ( tofind amax ormin )25 - 2x =0 so25 =2x and x = 25/2 = 12.......
a board hat hat is 25 feet long is to be but into two pieces where should it be cut so that the product of the two lengths is a maximum? not sure how to do this

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Usually Area questions like this become squares, Spheres, etc, with lots of symmetry in the two dimensions.

Let the first length be x feet
Then the second cut will then be ( 25 -x)
The area will be length 1 by length 2
or the product is ( length 1) by (length 2)

so A = Prod = x(25-x) = 25x - x^2...
This is a parabola, opening downwards so the derivative will give a maximum.

and differentiating this product function :
A' =P' = d(25x - x^2) / dx
= 25 - 2x
and setting this derivative to zero : ( to find a max or min )
25 - 2x = 0
so 25 = 2x and x = 25/2 = 12.5

The first length is x = 12.5 and the second length is (25 - 12.5) = 12.5 as well

so the cuts are at 12.5, and 12.5<---- answer for the two cut lengths

and the product is P = 12.5^2 = 156.25<--- answer for the maximum product
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