Posted by **Erica** on Thursday, October 11, 2012 at 7:19pm.

A fence 8 feet tall stands on level ground and runs parallel to a tall building. If the fence is 1 foot from the building, find the shortest ladder that will extend from the ground over the fence to the wall of the building.

- calculus -
**Reiny**, Thursday, October 11, 2012 at 9:27pm
I assume you made a sketch.

let the distance from the fence to the foot of the ladder be x

let the height of the ladder up the wall be h

by ratios:

8/x = h/(1+x)

hx = 8(1+x

h = (8+8x)/x

let the length of the ladder be L

L^2 = h^2 + (1+x)^2

= (8+8x)^2/x^2 + (1+x)^2

= (64 + 128x + 64x^2)/x^2 + 1 + 2x + x^2

= 64/x^2 + 128/x + 64 + 1 + 2x + x^2

2L dL/dx = -128/x^3 - 128/x^2 + 2 + 2x

= 0 for a min of L

times -x^3

128 + 128x - 2x^3 - 2x^4 = 0

128(1+x) - 2x^3(1+x) = 0

(1+x)(2x^3 - 128) = 0

x = -1 , which makes no sense

or

2x^3 - 128 = 0

x^3 = 64

x = 4

So L^2 = (8 + 32)^2 /4 + (1+4)^2

= 400 + 25

L^2 = 425

L = √425 = 5√17 = appr 20.6 ft

check my arithmetic.

- calculus -
**Anonymous**, Monday, December 8, 2014 at 6:05pm
This solution is correct but isn't the last step wrong because isn't it (8+32)^2/4^2 instead of (8+32)^4/4?

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