Width of Shelf Spaces / Partial Solution

[Guest]

A bookcase is to be constructed. The distance from the top of the lowest shelf to the bottom of the top shelf is 4.5 feet. The bookcase is to be constructed in a way so 4 other shelves of 1 inch thickness will have the book space diminish 1 inch from the bottom to the top. Determine spaces between shelves.

Suggestion with problem:

Let X = amount of inches between the 2 lowest shelves.

X + 1 = amount of inches to top of 2nd shelf.

X + 1 + (X - 1) + 1 = amount of inches to top of 3rd shelf.

How do I continue ?

[Guest]

Keep going why did you give up.

To be awkward but probably more simple is to let the space between the top shelf ant the highest new shelf be X. Then every other space will be bigger so add on 1 inch each time....... and 4.5 feet is 53 inches

So.....X + X + 1 + X + 2 + X + 3 + X + 4 + 4 = 53

....and solve for X

[Guest]

X + X + 1 + X + 2 + X + 3 + X + 4 + 4 = 54 (You miscalculated).

5X + 14 = 54
5X + 14 - 14 = 54 - 14
5X = 40
X = 8

On the suggestion:

How would the next steps be written? Just curious. Thanks.

[Guest]

OK 3 and 4 are very close

How would the next steps be written? Just curious. Thanks

X is the smallest space, the question says add 1 to find each other as we did in the equation

[Guest]

Here is a diagram:

---------------------------------------

---------------------------------------

-------------------------------------- Height 4.5 ft. = 54 in.
X + 1 + (X-1) + 1
-------------------------------------
X + 1
-------------------------------------
X
-------------------------------------

I still can't solve.

[Guest]

My diagram looks different.....................

-----------------Top of top shelf
=================thickness of top shelf
-----------------bottom of top shelf
X=8 inches space
-------------------top of next shelf
============= 1 inch thick
-------------------bottom of shelf
8 + 1 = 9 inches space
-------------------top of next shelf
============= 1 inch thick
-------------------bottom of shelf
8 + 2 = 10 inches space
-------------------top of next shelf
============= 1 inch thick
-------------------bottom of shelf
8 + 3 = 11 inches space
-------------------top of next shelf
============= 1 inch thick
-------------------bottom of shelf
8 + 4 = 12 inches space
-------------------top of bottom shelf
============= 1 inch thick bottom shelf
-------------------bottom of bottom shelf


The spaces diminish 1 inch each from bottom to top
Distance from top of bottom shelf to bottom of top shelf is 54 inches
Shelves are 1 inch thick

[Guest]

I don't understand.

[Guest]

I will space it out a bit more.....It is only a "typed" illustration to resemble the shelves one above the other with a space in between



-----------------Top of top shelf
================= the thickness of top shelf....this is meant to be the very top shelf of the bookcase.
-----------------bottom of top shelf .........We measure the 54 inches height from the top of the bottom shelf to here...*XXXXXXX*


X=8 inches space.......We let X = the space between the top shelf and the top new shelf that was added....this is that space and it worked out at 8 inches.


-------------------top of next shelf
============= 1 inch thick.....................this is the highest shelf that was added
-------------------bottom of shelf




8 + 1 = 9 inches space.........and the space is now 1 inch more than the space above so it is 9 inches




-------------------top of next shelf
============= 1 inch thick...........this is the next shelf down
-------------------bottom of shelf





8 + 2 = 10 inches space .......and the space is now 10 inches





-------------------top of next shelf
============= 1 inch thick ........this is the next shelf down
-------------------bottom of shelf




8 + 3 = 11 inches space .....and space is 11 inches






-------------------top of next shelf
============= 1 inch thick...........this is the next shelf down....the lowest shelf that was added
-------------------bottom of shelf




8 + 4 = 12 inches space ....and space is now 12 inches




-------------------top of bottom shelf.......we measure the 54 inches height from here to bottom of the very top shelf...*XXXXXXX*
============= thickness of bottom shelf......This is the very bottom shelf of the bookcase
-------------------bottom of bottom shelf


....................................

The spaces between shelves diminish 1 inch each from bottom to top.....so the spaces are ... 12 , 11, 10, 9 and 8 inches

Distance from top of bottom shelf to bottom of top shelf is 54 inches .....If you add up all the spaces and each shelf thickness you get 54
Shelves are 1 inch thick.......and you can see the shelves are 1 inch thick.

[Guest]

Thanks. I understand now.

[Guest]

One comment:

" If you add up all the spaces and each shelf thickness you get 54 " 8 + 9 + 10 + 11 + 12 + 6 = 56. The problem doesn't state thickness of the bottom and top shelves, only the 4 added. 54 would be the total then.

[Guest]

This was the original question...........
A bookcase is to be constructed. The distance from the top of the lowest shelf to the bottom of the top shelf is 4.5 feet. The bookcase is to be constructed in a way so 4 other shelves of 1 inch thickness will have the book space diminish 1 inch from the bottom to the top. Determine spaces between shelves.

You have to follow the information given in the original question.......

"The distance from the top of the lowest shelf to the bottom of the top shelf is 4.5 feet." ......... This is the only dimension given for the height of the bookcase and the new shelves that were added were added between the top of the original bottom shelf and the bottom of the original top shelf. The thickness of the original top or original bottom shelf does not matter as it is outside the dimensions given. There were to be 4 shelves added inside this dimension with the space between each added shelf getting 1 inch smaller as we go up each shelf and there needed to be a space from the top of the original bottom shelf to the bottom of the first lowest added shelf (the largest space), and similarly there needed to be a space between the top of the top added shelf and the bottom of the top original shelf (the smallest space and we let this equal "X"). And there were 2 other shelves giving 3 more spaces. And each shelf added had a thickness of 1 inch. This all adds up to 54 inches. So we put all these together to form an equation in terms of "X" and solved for "X" to find the dimension of the top space.......then we added one inch for each space to find the dimensions of the rest of the spaces.

[Guest]

I understand now. Thanks again.

One more question:

X + X + 1 + X + 2 + X + 3 + X + 4 + 4 = 53. You set up the equation from the top (said it would be easier).

The suggestion started from the bottom:

Let X = amount of inches between the 2 lowest shelves.

X + 1 = amount of inches to top of 2nd shelf.

X + 1 + (X - 1) + 1 = amount of inches to top of 3rd shelf.

My question: How would this way continue? Just curious.

[Guest]

If you are going to start from bottom and work up then you have to subtract 1 inch each time
Let X = bottom space
X + X-1 + X-2 + X-3 + X-4 + 4 = 54
5X - 6 = 54
5X = 60
X = 12 for bottom space as before........
The spaces between shelves diminish 1 inch each from bottom to top.....so the spaces are ... 12 , 11, 10, 9 and 8 inches ...as before

You mustn't have read through or understand the previous calculations or you would have known how to do this.

[Guest]

I was confused on the suggestion.