Square Box Size

[Guest]

In the Old West an express company delivers two boxes. The driver is to charge $5 per cubical foot as per instructions on the bill. The customer says the custom is to pay per running foot. The driver measured the length of the boxes and charges $5 per running foot. The boxes are square and one is half the height of the other. The driver placed the boxes together, measured the combined length and discovered less than .001 cent difference in either $5 per cubical foot or $5 per running foot.

Calculate sizes of the boxes. Thank you.

[Guest]

The box has square ends.....square section throughout.
The size of the square end is... (Root 5) x (Root 5) feet
............I can't type square root signs...??
........The boxes can be any length or height...........

.....................Simple...............

[Guest]

...........Sorry.........Root 1 ....which equals 1............
Length doesn't matter..????????

..................Simple......typing error..............

[Guest]

According to answers given, larger box is over 10,000 square inches and smaller box is over 5,000 square inches. Cost was less than $10. I don't understand how the certain answer given is "the answer" since you indicated the boxes could be any length or height. Thanks again.

[Guest]

Well, My simple answer is...........

The volume of a box L x B x H, length x breadth x height.
Another way of saying this is L x B is the area of one end and H is the height.
So volume is the area of the end multiplied by the height.....( L x B ) x H
And in the case of the question the end is square so volume is ( B x B ) x H if we say that L = B.

If the box has an end area of 1 square foot then its volume will be 1 x H = H square feet. So volume for costing is H cubic feet.

And for the running length costing, the height of the box is H so the length for costing is H feet.

The 2 methods of costing are... 5 dollars per running foot length or 5 dollars per cubic foot............

So if the end area is 1 square foot the cost will be the same for both boxes and the cost will be 5 x H = 5H dollars.

So it doesnt matter what length the box is, the cost by volume and cost by length will be the same if the end area is 1 square foot.

In the case of this question the end is square so the length of the sides of the square end will be ( Root 1 ) feet. which is 1 foot. because 1 x 1 = 1.

The question does not give any absolute value for actual cost so we can only work on cost per foot or per cubic foot....

.....................Simple Solution...........until I am proved wrong mathematically..............
PS........Does no one do Maths on this forum, do they just post questions..........

[Guest]

Sorry.......still another typing error.....

This line in the above posting..........
If the box has an end area of 1 square foot then its volume will be 1 x H = H square feet. So volume for costing is H cubic feet.

.........should read cubic feet as below.........
If the box has an end area of 1 square foot then its volume will be 1 x H = H cubic feet. So volume for costing is H cubic feet.

..........................Simple.............................

[Guest]

Using formula you provided (l X B X H) with answer I have of total square inches, square feet, and cubic feet, how would I calculate length, width and height of each box? Thanks again.

[Guest]

the question says they are square boxes...........does that mean they cubes......

L, B and H will all be the same..............??????

volume is cubic feet or cubic inches not square inches

[Guest]

One is half the height of the other. Thanks.

[Guest]

yes, probably wrongly described in the question, probably 2 cubes, one half the side length of the other, so one is quarter the volume of the other.
so their volume in cubic feet (not square feet) is equal to their length in feet.

[Guest]

Yet another typing error...............

In the previous post this line............
yes, probably wrongly described in the question, probably 2 cubes, one half the side length of the other, so one is quarter the volume of the other.

Should read........................

yes, probably wrongly described in the question, probably 2 cubes, one half the side length of the other, so one is one eighth the volume of the other.


.......................Simple......2 x 2 x 2 is 8............

[Guest]

Ok, we will state the facts we know from the the question.

Two boxes were delivered....Does it matter one, two, three or how many.

Two methods of charging.....$5 per cubic foot of volume......$5 per running foot of the lengths of the boxes added together.

The boxes are "Square".... does that mean square ends and whaterver length...????
Or does it mean square because all the corners are square. Is a rectangular box a square box...????
Or does it mean it is square because when you look at each face or side it looks square. In that case it is really a cube, all sides and faces are equal.

One box is half the height of the other. That implies it is standing on its end....its square end. So both are standing on their ends and one is half the height of the other.

In the case of the cubes for one to be half the height of the other, each side must be half the width of the other cube's side. So the large cube will be 8 times the volume of the small cube.

The cost is the same whether calculated by volume or by running length. we are told this.

We are not told the actual cost of this delivery, just the cost per foot running length, and the cost per cubic foot of volume. So we only have rates and no absolute values.

The previous posts dealt with the case of square end and the length did not matter, as long as the square end had an area of 1 square foot then the cost to deliver was the same whether calculated by running length or by volume.


This post deals with the case for cube boxes..............


Let the length of the side of the small cube box be L.
Then the length of the side of the large cube box is 2L

Volume of small cube is L^3.
Volume of large cube is 8L^3.
Total volume is 9L^3.
Total running length is 2L + L = 3L

So...... 9L^3 = 3L

3L^2 = 1

L = sqroot( 1/3 ) feet. = ( sqroot(3) ) / 3 feet.


Volume of small cube is ((sqroot(3)) / 3)^3

Volume of the large cube is 8 times this.

So total volume both cubes is 9 times it... = 9 x ((sqroot(3)) / 3)^3

The numbers are easy and simplifies to .... ( sqroot(3)) cubic feet.


The total running length is 2L + L = 3L = .... 3 x ( sqroot(3) ) / 3

Which simplifies to ......( sqroot(3)) feet


So the volume of the cubes is the same as the running length, so the cost will be the same no matter how it is calculated

And the size of the side of small cube is.... (1.732 / 3 ) x 12 = 6.928 inches x 6.928 inches x 6.928 inches.

And the size of the side of large cube is twice that ..... 13.856 inches x 13.856 inches x 13.856 inches.


....................Simple........................

[Guest]

I suppose I should work out the cost.......

Total length is (sqroot(3) feet at $5 per foot...

Equals........ 1.732 x 5

Equals..... $8.66 .......Equals $8 66 cents.

and this is same cost if worked out by volume..........

...................Simple................

[Guest]

I am not clear on the following parts of your solution:


So the large cube will be 8 times the volume of the small cube -Please show how you obtained.

3L^2 = 1- Please show how you obtained.


Thanks again.

[Guest]

OK, You query how I got the following......

So...... 9L^3 = 3L

3L^2 = 1

And you query how the large cube is 8 times the volume of the smaller.....

We are told the height of the small cube is half the larger one.
If it is a cube then all sides must be half the size of the larger one.
So that means the larger sides is twice the size of the smaller one.
so volume will be 2 x 2 x 2 = 8 times the smaller one.
But the size of the small cube sides is L so the large side size is 2L
So volume of small is ...... L x L x L = L^3
And volume of large is..... 2L x 2L x 2L is 8L^3
and 8 x L^3 is 8L^3 so 8 time larger than the smaller

For example..a foot is 12 inches......12 times bigger than the size of an inch
Similarly a square foot will be 12 x 12 = 144 times bigger in area than a square inch
and similarly a cubic foot will be 12 x 12 x 12 = 1728 times the volume of a cubic inch.

To get back to the question.... the total volume of the boxes delivered will be 8L^3 for the large one plus L^3 for the small one........This gives a total volume to be billed as 9L^3. .... And we are told the cost is the same as for the total length of the 2 boxes added together which is..... 2L + L = 3L. The cost per foot and per cubic foot is the same at $5 per foot or per cubic foot. So it does not matter how the bill is costed if the volume of the 2 boxes is the same number as the sum of the 2 lengths/heights.........

So...... 9L^3 = 3L
If I divide both sides of the equation by 3L.......

......then I get ....... 3L^2 = 1

and rearranging...... L^2 = 1/3

and................... L = sqroot( 1/3)

which equals........... L = sqroot (1) / sqroot (3) and multiply top and bottom by sqroot(3) to rationalise the surds

gives me..................L = sqroot (3) / 3. feet for size of small box and large box side will be twice that.

........Hope that explains it...............Simple..................

PS...........You have the question and from your previous posts you must have an answer..........
............Can you not publish your solution to let us all see how it is done....................???????
..........I don't have a problem with a solution...my problem is interpreting the question.......

[Guest]

All I have is the question and answer (same as your answer). I did not know how to work out the problem. Thanks again.