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
-
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
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