Rich is shipping a galvanized steel rod of negligible width in

Question

Rich is shipping a galvanized steel rod of negligible width in a rectangular container that measures 12 feet by 16 feet by 20 feet. What is the length, in feet, of the largest rod that he can fit in the container ?

(A) \(10\sqrt{2}\)
(B) 20
(C) 30
(D)\( 20\sqrt{2}\)
(E) \(30\sqrt{2}\)

(A)  10\sqrt{2} 
(B) 20
(C) 30
(D)  20\sqrt{2} 
(E)  30\sqrt{2}

Duuperman · Answer

The longest length will be the diagonal of the box. So you need to apply Pythagoras theorem twice to find out the lengths. First find the diagonal of the top surface and later find the diagonal of the box.

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IanStewart · Answer

You could use the three dimensional version of the Pythagorean Theorem: if a, b, and c are three mutually perpendicular lengths between two points in 3-dimensions (so for example, if they are the length, width, and height of a box), the straight line distance d between those points is given by

d^2 = a^2 + b^2 + c^2

Or you can use the ordinary Pythagorean Theorem twice. The base of our box measures 12 by 16, so the diagonal along the base is 20 (since this is just four times as big as a 3-4-5 triangle). So we can now make a right triangle using the diagonal along the base and the height of the box. We have a base diagonal of 20, and a height of 20, so our triangle is an isosceles right (45-45-90) triangle, and the hypotenuse is √2 times either side, so is 20√2.

fmcgee1 · Answer

Can I ask, how do we know what the base is? Like if I use the base as 12 x 16, and the height as 20, I will get a different answer than if I use the base as 20 x 12 and the height as 16.

IanStewart · Answer

It truly doesn't matter - I used 12 by 16 as the base because it was the easiest set of numbers to work with, but you'll get the same answer no matter how you orient the box.

billionaire999 · Answer

Diagonal of a rectangular box = \sqrt{H^2 + B^2 + L^2 }

= \sqrt{12x12 + 16x16 + 20x20 }
=\sqrt{800}
=20\sqrt{2}

Hope it helps all

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Rich is shipping a galvanized steel rod of negligible width in

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