Bend
What is the longest stick that can be carried horizontally along a
narrow corridor and around a right-angled bend?
Problem
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A long stick has to be carried horizontally along a narrow corridor and around a right-angled bend. The corridor is $a$ centimetres wide on one side of the bend and $b$ centimetres wide on the other side. Find, in terms of $a$ and $b$, the length of the longest stick that can be manoeuvered horizontally around the bend?
If the object is $2$ metres long and the two branches of the corridor are $65$ centimetres and $75$ centimetres respectively is it possible to manoeuvre the stick around the bend?
Getting Started
Work out the length of the straight line at an angle $\theta$ say
to one wall which just touches the inner corner in terms of $a,\ b$
and $\theta$. This length will change as $\theta$ changes and it
won't be possible to manoeuvre an object around the corner which is
longer then the minimum value of this length.
Student Solutions
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This solution comes from Andrei Lazanu ,Tudor Vianu National College, Bucharest, Romania.
I calculate the length of the stick in terms of $a$, $b$ and $\theta$. From the figure I observe that the length of the stick could be seen as the sum of two hypotenuses of two right-angled triangles. Its length is:
If $a=65 \text{ cm}$ and $b=75 \text{ cm}$ then $65^{2/3}+ 75^{2/3}=16.16623563 + 17.78446652=33.95070215$ and $33.951^{3/2}=197.8213407$ so an object of about $197 \text{ cm}$ could be manoeuvred around the bend but it is not possible to manoeuvre a $200 \text {cm}$ object around this bend.
Teachers' Resources
Some elementary calculus and trigonometry are required here and
some perseverance. You should expect the final answer to be an
expression symmetric in $a$ and $b$.