# A fence 4 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?

#( (x = h_0 cot(alpha)), (y = d_0 tan(alpha)), (l = h_0/sin(alpha) + d_0/cos(alpha)) )#

or

and

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8.32ft

Intending to use a Lagrange Multiplier , we are minimising

This is subject to constraint which comes from similar triangles that

so

using the constraint

ignoring the trivial solution we have

so ladder length

plot confirms authenticity of solution

The "proof" that this is a minimum comes from physical arguments. It is easy to imagine a ladder that has its base a distance

then

and

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The length of the shortest ladder that will reach from the ground over the fence to the wall of the building is ( 2\sqrt{13} ) feet.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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