| I leaned the ladder on the other side of the wall, and my choice
of letters is not consistent with the diagram in .0.
Let the wall be the y axis, the floor be the x axis, and let
the wall and floor meet at the origin. The equation of the
ladder if it intersects the x axis at (a,0) and the y axis at
(0,b) is:
ay + bx = ab
Since the length of the ladder is 8
a^2 + b^2 = 8^2 = 64
The cube of length 1 touches the line at the cube's corner at
(1,1). Plugging this into the equation of the line,
a + b = ab
Combining these:
(a + b)^2 = a^2 + 2ab + b^2
= (a^2 + b^2) + 2ab
= 64 + 2(a + b)
Let x = (a + b), then
x^2 = 64 + 2x
x^2 - 2x = 64
x^2 - 2x + 1 = 65
(x - 1)^2 = 65
x - 1 = sqrt(65), or x - 1 = -sqrt(65)
x = 1 + sqrt(65), or x = 1 - sqrt(65)
Since x = (a + b) is not negative, it must be the case that
a + b = ab = 1 + sqrt(65)
Then,
(a - b)^2 = a^2 - 2ab + b^2
= (a^2 + b^2) - 2ab
= 64 - 2(1 + sqrt(65))
= 62 - 2 sqrt(65)
(a - b) = + or - sqrt(62 - 2 sqrt(65))
Combining this with a + b = 1 + sqrt(65) gives two solutions,
(a,b) = (r,s) and (a,b) = (s,r), where
r = 1/2 * (1 + sqrt(65) + sqrt(62 - 2 sqrt(65))) (approx. 7.92)
s = 1/2 * (1 + sqrt(65) - sqrt(62 - 2 sqrt(65))) (approx. 1.14)
The height, b, of the ladder can be either r or s.
Dan
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