What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building ? Following will be considered when answering the problem. a) Find the equation to maximized or minimized. b) Finding the solution c) Showing that your solution is an absolute maximized or minimized. A fence 8 ft tall runs parallel to a tall building at a distance of 4 ft from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building ?

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Let's say the distance of the base of the ladder from the fence is x.  So `(x+4)/x = h/8` we get `h = (8(x+4))/x` .   The length of the ladder is `L=sqrt(((8(x+4))/x)^2 + (x+4)^2)` .   We can take the derivative with respect to x and set = 0 to...

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Let's say the distance of the base of the ladder from the fence is x.  So `(x+4)/x = h/8` we get `h = (8(x+4))/x` .   The length of the ladder is `L=sqrt(((8(x+4))/x)^2 + (x+4)^2)` .   We can take the derivative with respect to x and set = 0 to find the extrema.

`(dL)/(dx)=(2(64)(((x+4))/x)((x-(x+4))/(x^2)) + 2(x+4))/(2L)`

this is zero when `-(256(x+4))/x^3+(x+4) = 0`

Simplifying we get `(x+4)-256(x+4)x^3=(x+4)(256-x^3)=0`

`x=-4` or `x=root(3)(256)~~6.635`

This gives `h = sqrt((8+root(3)(256))^2 + (root(3)(256)+4)^2) ~~ 16.648` ft

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