Find the greatest number of boxes which can be packed in the crate

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bumblebee123

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Can anyone help me- I don't understand why the answer is 124 and not 125

Question: A crate is a cube with side 1 m. A box is a cuboid which is 40cm by 20cm by 10cm. Find the greatest number of boxes which can be packed in the crate.

So the volume of the crate must be 1m^3. And the cuboid volume must be 0.4 m x 0.2 m x 0.1 m = 0.008m^3.

to find out how many boxes fit in the crate I thought it would be: 1 / 0.008 = 125

however the answer is 124- why is this!?

any help would be highly appreciated :)
 
If you could reshape the boxes to fit, you could use volume. But you have to be able to actually fit them into the given dimensions, as they are. So work with lengths, not just volumes.
 
You need to be careful with just doing the division as you did. Suppose you had a box that is 4m by 2 m by 1m. So the volume is 8m3. Can you really put in 1 box whose volume is 8m by 1m by 1m? After all the volume of this box is 8m3 and 8m3 / 8m3 = 1.

BTW, I am not getting 124 nor 125
 
I do get 124. First I made a layer 80 cm deep x 100 x 100; then a block 20 cm deep x 80 x 100; then fill in the remaining 20 x 20 x 100 as full as possible, leaving a 20 x 20 x 20 space empty.
 
Jomo said:
You need to be careful with just doing the division as you did. Suppose you had a box that is 4m by 2 m by 1m. So the volume is 8m3. Can you really put in 1 box whose volume is 8m by 1m by 1m? After all the volume of this box is 8m3 and 8m3 / 8m3 = 1.

BTW, I am not getting 124 nor 125
Click to expand...

The answer in the book says 124 ( idk how they got it either! )
 
bumblebee, do you understand why you can't just go by volume (did you understand my example above?)
And yes, Dr Peterson is correct with 124. Take your time and consider what is said carefully.
 
Jomo said:
bumblebee, do you understand why you can't just go by volume (did you understand my example above?)
And yes, Dr Peterson is correct with 124. Take your time and consider what is said carefully.
Click to expand...

yes, I think I understand it- it also makes sense that the cubes have to fit as what was said so I can't use volume to mould them. I'm just confused as to how I would now figure the answer out.
 
bumblebee123 said:
yes, I think I understand it- it also makes sense that the cubes have to fit as what was said so I can't use volume to mould them. I'm just confused as to how I would now figure the answer out.
Click to expand...
Try to understand how Dr Peterson chose to arrange it. I thought that getting the entire base filled would yield the max number of cuboids but I was wrong as I had much too much unused space (volume) at the top.
 
Jomo said:
Try to understand how Dr Peterson chose to arrange it. I thought that getting the entire base filled would yield the max number of cuboids but I was wrong as I had much too much unused space (volume) at the top.
Click to expand...

where have the figures come from? why is it 80 cm and why is it x 100 x 100? sorry I'm a bit confused
 
Have you tried??? That's where the numbers came from: trying!

Do you imagine that I used some formula I know to calculate it? No, I just thought about how I could pack a bunch of boxes in one orientation. We want to cover a "floor" 100 by 100 cm with a layer of boxes. Since 40 does not divide 100 evenly, I imagine standing the boxes up on their 20 x 10 ends. That way, I can make 5 rows (5 x 20 = 100) and 10 columns (10 x 10 = 100). This makes a layer 40 cm deep. How many will be in that layer? How many such layers can I make?

And so on. No magic formulas; just thinking about how to do it, step by step. You might find a different way; if so, tell us what you've done, and we can suggest what to do next. But you won't learn to solve problems without trying.

By the way, if you have trouble visualizing in three dimensions, notice that my thinking is mostly two dimensions at a time. You can sketch the 100x100 floor and how to fill it with rectangles, for example.
 
Dr.Peterson said:
Have you tried??? That's where the numbers came from: trying!

Do you imagine that I used some formula I know to calculate it? No, I just thought about how I could pack a bunch of boxes in one orientation. We want to cover a "floor" 100 by 100 cm with a layer of boxes. Since 40 does not divide 100 evenly, I imagine standing the boxes up on their 20 x 10 ends. That way, I can make 5 rows (5 x 20 = 100) and 10 columns (10 x 10 = 100). This makes a layer 40 cm deep. How many will be in that layer? How many such layers can I make?

And so on. No magic formulas; just thinking about how to do it, step by step. You might find a different way; if so, tell us what you've done, and we can suggest what to do next. But you won't learn to solve problems without trying.

By the way, if you have trouble visualizing in three dimensions, notice that my thinking is mostly two dimensions at a time. You can sketch the 100x100 floor and how to fill it with rectangles, for example.
Click to expand...


yes, I tried for a very long time. it's just very difficult for me to visualize. so there are five 20cm wide boxes, which go 10cm back
( this goes ten boxes back), and are 40cm tall. 100 / 40 = 2.5, so it can only go up two layers

each layer = 5 x 10 boxes = 50 boxes

there are two layers so 50 x 2 = 100 boxes
 
So you've used 100 boxes in the first two layers; now I think you're making a layer "lying down" so the 40 and 10 cm are horizontal, and the layer is 20 cm high. You have a row of 10 boxes; but how many of these rows are there?

Did you make a picture of this layer? It will help a lot.
 
Please trust Dr Peterson's result of 124. If you get 100, good! But you can do better. Just keep trying and trying. Who knows maybe you will get 125 which must be the max as you have the constraint on the volume. This is why math opens up your mind! I am not by any means a great math person but studying math taught me to think and think differently.
 
Jomo said:
Please trust Dr Peterson's result of 124. If you get 100, good! But you can do better. Just keep trying and trying. Who knows maybe you will get 125 which must be the max as you have the constraint on the volume. This is why math opens up your mind! I am not by any means a great math person but studying math taught me to think and think differently.
Click to expand...

I don't doubt dr. peterson has the right answer! I know he knows what he's doing ( much more than I do haha ). My brain's just a bit fried trying to understand ?
 
11844

( I hope you guys can read everything alright )

so in all three layer = 100 + 25 = 125 boxes
 
Well, we can't have 2.5 boxes in a row; so the 25 you show isn't right.

To show you what I meant previously about using 2-d pictures to keep things simple, here are pictures (top view) of the first two layers (each 40 cm deep)
FMH115624a.png
and the final layer (20 cm deep), which consists of two groups of boxes and an empty space
FMH115624b.png
Of course, there are other ways; I based this arrangement on what you have said.

Can you see how to do the counting now?
 
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