Question

A fence 6 feet tall runs parallel to a tall building at a distance of 2 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?

Answer 1

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
6/x = y/(x+2)
xy = 6x+12
y = (6x+12)/x

let the length of the ladder be L
L^2 = (x+2)^2 + y^2
= (x+2)^2 + [(6x+12)/x]^2

2L dL/dx = 2(x+2) + 2[(6x+12)/x] (x(6) - (6x+12))/x^2
= 0 for a min of L

x+2 + (6x+12)(-12)/x^3 = 0
.. ..
x^4 + 2x^3 - 72x - 144 = 0
x^3(x+2) - 72(x+2) = 0
(x^3 - 72)(x+2) = 0
x = -2 , not possible
or
x = 72^(1/3) , (which is the cuberoot of 72)

sub back into L^2 = ....

Answer 2

THANKS YOU Reiny!!! You are a great help. Turns out my "y" was wrong but I did get the "x"

Answer 3

Did you get a shortest ladder of 9.81 ft ?

Answer 4

I actually got 10.811 ft