Posted by Anonymous on Friday, October 28, 2011 at 9:20pm.
A fence 4 feet tall runs parallel to a tall building at a distance of 6 feet 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?

Calculus  Reiny, Friday, October 28, 2011 at 11:30pm
Make a diagram
let the foot of the ladder be x ft from the fence
let the ladder reach y ft above the ground
I see similar triangle so set up a ratio
4/x = y/(x+6)
xy = 4x+24
y = (4x+24)/x
let the length of the ladder be L
L^2 = (x+6)^2 + y^2
= (x+6)^2 + [(4x+24)/x]^2
2L dL/dx = 2(x+6) + 2[(4x+24)/x] (x(4)  (4x+24))/x^2
= 2(x+6)  8(x+6)(24)/x^3
= 0 for a min of L
2(x+6)  192(x+6)/x^3 = 0
times x^3
2x^3(x+6)  192(x+6) = 0
2(x+6)(x^3  96) = 0
x = 6 , which makes no sense
or
x = 96^(1/3) , (which is the cuberoot of 96)
x = 4.57886
sub back into L^2 = 197.3
L = 14.05 m
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