Using Trigonometry formulae,
the length of the longest item that can be carried horizontally around the corner is 19.60 feet.
Let's try to draw it and try to solve with a diagram we're moving into a new apartment. We see that the right corner is 8 feet wide, but then that's right there is 6 feet wide and we're going to have some ladder or a long item required here .
Creating some angle θ and let b feet be the distance parallel to 8 feet and a be the distance parallel to 6 feet and then one portion we'll call d, other one portion e . So, the total length (L) = d + e
In this case, sinθ = b/e,
because that's the opposite over the hypotenuse, => e sinθ = b
and d cosθ = a
We know that e = b cosecθ and d = a secθ
so the length(L) = a secθ + b cosecθ In order to maximize length , we want to take the derivative with respect to θ we get,
dL/dθ = - b cotθ cosecθ + a secθ tanθ
now, dL/dθ = 0
=> - b cotθ cosecθ+ a secθ tanθ = 0
=>b cotθ cosecθ = a secθ tanθ
=>tan³θ = b/a
=> tanθ = (b/a )⁰·³³
plugging value of b= 6 and a = 8
tanθ = (6/8)⁰·³³
=> tanθ = 0.9085
so, θ = tan⁻¹(0.9085)
=> θ = 42.2550
plug the value of θ in e and d
we get , e = 6 cosec(42.2550) = 8.8 d = 8 sec(42.2550) = 10.80
then, L = d + e
=> L = 8.8 + 10.80 = 19.60
Hence, the required length is 19.60 feet .
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