What is the minimum perimeter?

Geometry Level 2

Consider all rectangles with integer side lengths and an area of 72 m 2 72m^2 . What is the minimum possible perimeter?


The answer is 34.

Thomas Lowry by Thomas Lowry , 22, Canada

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3 solutions

Mohammad Khaza
Jul 5, 2017

here it is,

72 =8 x 9

o, the perimeter is ={(8 + 9) x 2} 

                             =34 meter
2 Helpful 0 Interesting 0 Brilliant 0 Confused

F i n d i n g t h e m i n i m u m p e r i m e t e r m e a n s t h a t t h e d i f f e r e n c e b e t w e e n t h e s i d e l e n g t h s s h o u l d b e t h e s m a l l e s t . Finding\ the\ minimum\ perimeter\ means\ that\ the\ difference\ between\ the\ side\ lengths\ should\ be\ the\ smallest.

For 72 72 the minimum difference is 1 by multiplying 8 into 9 which can be found out by trial and error. Therefore the minimum perimeter is 34 m...... 2 2 x ( 8 + 9 ) (8 + 9)

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Peter Michael
Jun 12, 2017

Min means all sides

Use square roots to find the sides

Multiply by four

0 Helpful 0 Interesting 0 Brilliant 0 Confused

@Thomas Lowry Wouldn't the answer be 4 72 4 \sqrt{72} ? Are we assuming that the sides must be an integer?

Calvin Lin Staff - 3 years, 11 months ago

If you use derivatives to find the answer, you get a function of the perimeter: P(a) = 2 (a + 72/a), which has it's minimum at a = 6sqrt(2), yet the answer needs to be an integer ( my answer would be 24sqrt(2) ). What am I missing?

Jan Chomiak - 3 years, 11 months ago

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Thanks. I have edited the problem for clarity.

Calvin Lin Staff - 3 years, 11 months ago

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