G-7 Thick Trick

Geometry Level pending

A closed rectangular box is of uniform thickness x x inches. The box has outer dimensions 6 inches, 4 inches, 3 inches and it has an inside volume of 30 cubic inches. Find x x .


The answer is 0.5.

Brian Dela Torre by Brian Dela Torre , 21, Philippines

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2 solutions

Zeeshan Ali
Feb 6, 2016

I just came up with a guess that

( 6 − 2 i ) ( 4 − 2 i ) ( 3 − 2 i ) = 30 = 5 × 3 × 2 = ( 6 − 1 ) × ( 4 − 1 ) × ( 3 − 1 ) (6-2i)(4-2i)(3-2i)=30=5 \times 3 \times 2=(6-1) \times (4-1) \times (3-1)

⟹ t h a t 2 i = 1 , o r i = 0.5 : ) \implies \, that \, 2i=1 \,, or \, i=0.5 \, :)

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A fast method way of solving thanks.

Brian Dela Torre - 5 years, 4 months ago
Brian Dela Torre
Feb 4, 2016

Volume = l w h lwh

Inside volume = 30 i n 2 30 in^{2}

Inside volume = ( l − 2 x ) ( w − 2 x ) ( 3 − 2 x ) (l - 2x)(w - 2x)(3 - 2x)

30 30 = 6 − 2 x ) ( 4 − 2 x ) ( 3 − 2 x ) 6 - 2x)(4 - 2x)(3 - 2x)

8 x 3 − 52 x 2 + 108 x − 42 = 0 8 x^{3} - 52 x^{2} + 108x - 42 = 0

( 2 x − 1 ) ( 8 x 2 − 48 x + 84 ) = 0 (2x - 1)(8 x^{2} -48x + 84) = 0

Since 2 x − 1 2x -1 is a factor of 8 x 3 − 52 x 2 + 108 x − 42 = 0 8 x^{3} - 52 x^{2} + 108x - 42 = 0 then 2 x = 1 , x = 1 2 2x =1, x = \frac{1}{2}

Hence the answer is 1 2 \boxed{\frac{1}{2}}

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