Ladder over the fence An 8-foot-tall fence seperates Larry\'s yard from Evan\'s

Question

Ladder over the fence An 8-foot-tall fence seperates Larry's yard from Evan's yard. The fence is 3 feet from Larry's house and runs parallel to the side of his house. Larry wants to paint his house and needs to position a ladder extending from Evan's yard over the fence to his house. Assuming the vertical wall of Larry's house is less than 30 feet tall and Evan's yard is at least 20 feet long what is the length of the shortest ladder (rounded up to the nearest foot) whose base is in Evan's yard, clears the fence and reaches Larry's house? (problem Solving)

Explanation / Answer

derivation: ???? ........../| .....y./..| L..../?..| ..../|.....| x/..|.h..| /?..|.....| ...... d let ? be the angle the ladder makes with the ground & L ft be the length of ladder = x+y = (h/sin?) + (d/cos?) dL/dt = h cos?/sin^2 ? - dsin?/cos^2 ? = 0 for minima putting under common denominator, numerator hcos^3 ? - dsin^3 ? = 0 tan^3 ? = h/d tan? = (h/d)^(1/3) this means we have compressed the scale of the original drawing (taking its cube root), so L also gets cmpressed to L^(1/3) applying pythagoras' theorem, L^(2/3) = h^(2/3) + d^(2/3) L = ( (h^(2/3) + d^(2/3) )^(3/2) , i.e. L = ( (8^(2/3) + 3^(2/3) )^(3/2)

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