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