SOLUTION: A cubic box has inside dimensions of 10 by 10 by 10 inches. What is the longest pole of diameter 1 inch that will fit inside?

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Question 1075820: A cubic box has inside dimensions of 10 by 10 by 10 inches.
What is the longest pole of diameter 1 inch that will fit inside?
Answer by ikleyn(52799) About Me  (Show Source):
You can put this solution on YOUR website!
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The answer and the explanations to the answer.


The long (the longest) 3D diagonal is sqrt%2810%5E2+%2B+10%5E2+%2B+10%5E2%29 = 10%2Asqrt%283%29 inches long.

We should take off (cut off) some parts at the ends to fit the cylinder inside.
These parts are of the length of sqrt%282%29%2F2 inches from each end.

So the final length is 10%2Asqrt%283%29+-+sqrt%282%29 inches.

Answer. The length of the cylinder is 10%2Asqrt%283%29+-+sqrt%282%29 inches.


The solution 

1.  Make a sketch of the vertical section of the cube by the vertical plane through the 3D diagonal.

    In the section you have a rectangle with the base of 2%2Asqrt%2810%29 inches long and the height of 10 inches long.

    The diagonal of this rectangle is exactly the 3D diagonal of the cube and has the length of 10%2Asqrt%283%29 inches.


2.  The slope of the diagonal to the base is measured by the angle alpha that the diagonal makes with the base.

    You have tan%28alpha%29 = 10%2F%2810%2Asqrt%282%29%29 = 1%2Fsqrt%282%29 = sqrt%282%29%2F2.


3.  Plot the contour of the cylinder, which makes the strip of the wide 1 in the direction perpendicular to the diagonal.

    In the sketch of the vertical section, plot the straight segments that represent the bases of the cylinder.

    You will see small right-angled triangles that show which parts of the strip should be removed.



4.  The shortest legs of these triangles are o.5 inches long (each).
    The shortest legs are opposite to the angle alpha.

    Hence, the longest legs of these triangles are 0.5%2Ftan%28alpha%29 = 0.5%2F%28%28sqrt%282%29%2F2%29%29 = 1%2Fsqrt%282%29 = sqrt%282%29%2F2 inches long (each).



    It gives 10%2Asqrt%283%29+-+sqrt%282%29 inches for the length of the cylinder.

Solved.