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?

Answer 1

#8.32388# feet

Fence height = #h_0# Fence distance = #d_0# Ladder length = #l#
#l cos(alpha)= x + d_0# #l sin(alpha) = y + h_0# #(y+h_0)/h_0 = (x+d_0)/x# Thales of Miletus
Solving for #x,y,l# we have

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

Here #l(alpha)# so for stationary values determination we do
#(dl)/(d alpha) = -h_0 cos(alpha)/sin^2(alpha)+ d_0 sin(alpha)/cos^2(alpha) = 0#

or

#d_0sin^3(alpha)-h_0cos^3(alpha)=0 -> tan(alpha) = (h_0/d_0)^{1/3}# #alpha = arctan[(h_0/d_0)^{1/3}] = 0.899908#

and

#l(0.899908)=8.32388# feet
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Answer 2

8.32ft

Intending to use a Lagrange Multiplier , we are minimising #s^2 = x^2 + y^2 = f(x,y)# where s is the length of the ladder.

This is subject to constraint which comes from similar triangles that #y/x = 4 / (x-2) \implies yx - 2y - 4x = 0 = g(x,y)#

so #nabla f = lambda nabla g = implies #

#<2x, 2y > = lambda < y - 4, x-2>#

#\implies x/(y-4) = y / (x-2) qquad star#

using the constraint #y/x = 4 / (x-2) \implies y = (4x)/(x-2)# and subbing this into #star# :

#\implies x/( (4x)/(x-2)-4) = ((4x)/(x-2)) / (x-2) #

#\implies (x(x-2))/( 4x-4(x-2)) = (4x) / (x-2)^2 #

#\implies x(x-2)^3 = 32x#

#\implies x((x-2)^3 - 32) = 0#

ignoring the trivial solution we have

# (x-2)^3 = 32#

#x = 2 + 32^{1/3} = 5.175#

#y = 6.519#

so ladder length #s = sqrt{5.175^2+6.519^2} = 8.32 ft#

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 #epsilon# beyond the fence so that the similar triangles give

#y/(2 + epsilon) = 4 / epsilon |implies y = 8/epsilon + 4#

then #lim_{epsilon to oo} y = lim_{epsilon to oo} 8/epsilon + 4 = 4#

and #lim_{epsilon to 0} y = lim_{epsilon to 0} 8/epsilon + 4 = oo#

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Answer 3

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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Answer from HIX Tutor

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.

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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