A 3' cube sits on level ground against a vertical wall. A 12' ladder on the same ground leans against the wall such that it touches the top edge of the box.
How far from the wall must the foot of the ladder be, if it is to reach maximum height whilst meeting the foregoing conditions?
Let Pythagorus sleep.
The 12ft ladder meets the ground at angle a.
The edge of the 3ft box divides the ladder into 2 lengths whereby:
12 = 3/sin a + 3/cos a
4 sin a cos a = cos a + sin a
2 sin 2a = cos a + sin a
4 sin^2 2a = cos^2 a + sin^2 a + 2 sin a cos a
4 sin^2 2a = 1 + sin 2a
4 sin^2 2a - sin 2a - 1 = 0
sin 2a = (1 ± (1 + 16)^½)/8
a must be positive therefore the 2 values of 2a are 39.82 and 140.18 degrees.
For maximum height the distance of foot of ladder from the wall = 12 cos (140.18/2) = 4.09
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