# Calculus

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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 -

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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