A fold is formed on a #20 cm × 30 cm# rectangular sheet of paper running from the short side to the long side by placing a corner over the long side. Find the minimum possible length of the fold?

Answer 1

# L = 15sqrt(3) ~~ 25.98 \ cm#

Let us set up the following variables:

{

(L, "Length of the fold", cm),
(x, "DE", cm),
(theta, "Angle " hat(DGE), "radians")

:} #

We are given that #AD=30 cm" and "AB=20 cm#. Our aim is to find the minimum length #L# of the fold.
By trigonometry for #triangle DEG# we have:
# sin hat(DGE) = (DE)/(EG) => sin theta = x/L # ..... [A]
When folded the point #D# must touch the side #BC#
# hat(HFG) = hat (FGD) = 2theta#

Now:

# hat(HFG) + hat(GFE) + hat(EFC) = pi # # :. 2theta + pi/2 + hat(EFC) = pi # # :. hat(EFC) = pi/2 - 2theta #
By trigonometry for #triangle CEF# we have:
# sin hat(EFC) = (EC)/(EF ) # # :. sin (pi/2 - 2theta) = (CD-DE)/(EF ) # # :. sin (pi/2 - 2theta) = (20-x)/(x) #

Using the sum of angle formula:

# sin(A-B) -= sinAcosB - cosAsinB #

we have:

# sin(pi/2)cos(2theta) - cos(pi/2)sin(2theta) = (20-x)/(x) # # :. cos(2theta) = 20/x - 1 #

Using the identity:

# cos 2A -= 1-2sin^2A #

we have:

# 1-2sin^2theta = 20/x - 1 #
Using #Eq [A]# this becomes:
# \ \ \ \ \ 1-2(x/L)^2 = 20/x - 1 # # :. 2 -2 x^2/L^2 = 20/x # # :. x^2/L^2 = 1-10/x # # :. x^2/L^2 = (x-10)/x # # :. L^2/x^2 = x/(x-10) # # :. L^2 = x^3/(x-10) #
As #L# changes then #theta# and #x# change accordingly, we know have #L^2# as a function of #x# alone, and minimizing #L# is the same as minimizing #L^2#, so we seek a critical point of #L^2#
Differentiating wrt #x# by applying the quotient rule we have:
# d/dx L^2 = { (x-10)(d/dx(x^3)) - (d/dx(x-10))(x^3) } / (x-10)^2 # # \ \ \ \ \ \ \ \ \ \ = { (x-10)(3x^2) - (1)(x^3) } / (x-10)^2 # # \ \ \ \ \ \ \ \ \ \ = x^2{ 3(x-10) - x } / (x-10)^2 # # \ \ \ \ \ \ \ \ \ \ = x^2{ 3x-30 - x } / (x-10)^2 # # \ \ \ \ \ \ \ \ \ \ = x^2{ 2x-30 } / (x-10)^2 #

At a critical point this derivative vanishes, and so:

# x^2{ 2x-30 } / (x-10)^2 = 0 # # x^2(2x-30 ) = 0 # # x=0,15 #
# x =0 # result in no fold and so we required #x=15#, with this value of #x# we have:
# L^2 = 15^3/(15-10) # # \ \ \ \ = 3375/5 # # \ \ \ \ = 675 #
# :. L = 15sqrt(3) ~~ 25.98 \ cm#
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Answer 2

The minimum possible length of the fold on a 20 cm × 30 cm rectangular sheet of paper formed by placing a corner over the long side is 25 cm.

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