another math problem?

Question

A bookcase is to have four shelves as shown. The height of the bookcase is to be 2 feet more than the width, and only 20 feet of lumber is available. What should be the width and height of the bookcase? Use an equation to solve the problem.

yes it does have a top shelf

Answer 1

How many shelves?

Here's what you need to do:

(s+2)*w + 2*(w+2) = 20

sw + 2w + 2w + 4 = 20
sw + 4w + 4 = 20
sw + 4w = 16

(s+4)w = 16
w = 16 / (s + 4)
h = w + 2

where s is the number of shelves excluding the bottom and top.

Answer 2

OK. Considering that the bottom shelf is also the bottom side of the bookcase (no picture was present) and assuming that the remaining 3 shelves are also to be made from the 20 ft of lumber, then the following solution should work.

First realize that the bookcase is to be 2 ft more than the width. Thus if the width is x, then the height is x + 2. This creates a perimeter of:

P = 2x + 2(x+2)

Now, there are three more shelves which also must be a width of x, so we get 2x + 3x = 5x. Now we just add them all together as:

5x + 2(x + 2) = 20

and just solve for "x".

I get x = 16/7 ft or about 2.29 ft for the width and 4.29 ft for the height.

Answer 3

If the bookcase has four shelveds that mean you need 3 pieces in the middle plus one top piece and one bottom piece, which means you need 5 pieces in width and two pieces for sides. so lets assume L for Length and W for Width
2L + 5W = 20
also your question said L = W+2 subsitute this
2 (W+2) + 5W = 20
2W + 4 + 5W = 20
7W + 4 = 20
7W = 20-4
7W = 16
W = 16/7
W = 2.28
H = W + 2
H = 2.28 + 2
H = 4.24

Finally you have H = 4.28 and W = 2.28 to use 20 feet of Lumber

Answer 4

If you have a top shelf then you have 5 cross pieces that are x in width, and you have 2 support pieces that are x+2. So the equation you could write would be:
5(x) + 2(x+2)=20

Now you get the fun of solving for x to find the width of each shelf.

Hope this helps. Good Luck.

Answer 5

Solution #1 the naive solution

w = width of shelves in feet
h = hight of sides in feet

20 ft. = 4w + 2(w+2ft.)
20 ft. = 4w + 2w + 4ft.
16 ft = 6w
w = 16ft. / 6 = 8ft. / 3 = 2 2/3 ft. = 2 ft. 8 in.
h= 2 ft. 8 in. + 2 ft. = 4 ft. 8 in.

if this is a homework problem this is the solution you want.

Solution #2

if your lumber is one 20 ft. piece you have to
make 5 cuts to to get the 6 pieces, four shelves
and two sides. A saw removes part of the lumber
when it makes the cut, called the saw kerf. If the
kerf is 1/8 in. for 5 cuts you lose 5/8 in. of lumber.
therefor


20 ft. - 5/8 in. = 4w + 2(w+2ft.)
19 ft. 3/8 in. = 4w + 2w + 4ft.
15 ft 3/8 in. = 6w
w = (15 ft.x1/6) + (3/8 in.x1/6)
w = 2 ft. 6 in. + 1/16 in. = 2 ft. 6 1/16 in.
h = w + 2 ft. = 4 ft. 6 1/16 in.

these are the sizes if you want to build a book case from a 20 ft. piece of lumber. If you start with 2 - 10 ft. pieces you only make 4 cuts and have to recalculate. Starting (2 x 10 ft.) - (4 x 1/8 in.)
and going on as above.

Answer 6

assume 2 vertical pieces length = h
plus 4 horiz pieces (shelves) of length = w
total length = 2h + 4w = 20
h = w + 2 was given
sustitute in other equation
2(w+2) + 4w = 20
6w + 4 = 20
6w = 16
w = 16/6 = 2 and 2/3 = 2ft 8in
hence h = 4ft 8in

Answer 7

2 feet divided by 2 will be 2 pcs 20 feet.
Cut 2 pcs 1X5 for siding balance 30 feet divided by 5 pcs

Answer 8

Does your bookshelf "as shown" have a top? What I'm asking is... are there four or five cross pieces? You could add that information as an addendum to your question.

Aloha

Answer 9

how is the picture? 4 boxes in the shelves?

Answer 10

you need more lumber, and how many screws are to be used? and can we use a leveler top make sure the peices are level or do we just estimate with our naked eye