please i need quick help

a box of the dimensions 9 ,5, 8 m is filled with small matches box with the dimensions 0.12 , 0.3 , 0.16 m what is the largest number of matches boxes that can fit in the big box

Easy Math Editor

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Comments

The maximum number of match sticks that can be kept in a match box,

$n = \dfrac{\text{Volume of Matchbox}}{\text{Volume of one Matchstick}}$

$\implies n = \dfrac{9 \times \times 5 \times 8}{0.12 \times \times 0.3 \times 0.16} = \boxed{62,500~\text{matchsticks}}$

Ram Mohith - 2 years, 7 months ago

But maybe there are gaps...

I think the 5m side can't be used completely because 5 isn't divisible by 0.12, 0.3 or 0.16.

Maybe it's like putting 2×2 squares in a 5×5 square. You can only fit 4 in although there would be space for 6.

If we put all match boxes in the same orientation, we should put their 0.3m side along the 9m side to use it completely, then the 0.16m sides at the 8m side and then we have to align the 0.12m sides with the 5m side. This makes 30 matchboxes along the 9m side, 50 along the 8m side and 41 along the 5m side, with a gap of 0.08m, so we can't fit any more matchboxes in and have only $30 \cdot 50 \cdot 41 = 61500$ .

But maybe there is a more irregular arrangement to fit a few more in...

Henry U - 2 years, 7 months ago

I agree with @Henry U. @Ram Mohith, We have to actually fit them so we need to actually use some trial and error for which side has the most number of $k$ type matchsticks.

Mohammad Farhat - 2 years, 7 months ago

@Mohammad Farhat – Ok. Now check whether I am correct or not ?

The maximum number of matchsticks that an be placed in a matchbox is $\left \lfloor n \right \rfloor$ where $\left \lfloor . \right \rfloor$ denotes greatest integer function of $n$ . Suppose, if we get $n = 58.6$ then the maximum number of matchsticks will be $58$ .

Ram Mohith - 2 years, 7 months ago

@Ram Mohith – Ah! Yes!

Mohammad Farhat - 2 years, 7 months ago

you are right completely right the solution ram mohith provide is right if only it was a cube box and cube matches box

Ahmed Mahmoud - 2 years, 7 months ago

but still i can get more than 61500 i got 61980 by trying to fit the sides of the matches box by the big box and figure out what best fit that you can use and reuse the remaining margins but there still some possibilities I didn't have the time to try

Ahmed Mahmoud - 2 years, 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$