Question

An artist is creating a mosaic that cannot be larger than the space allotted which are 4 feet tall and 6 feet wide. The mosaic must be at least 3 feet tall and 5 feet wide. The tiles in the mosaic have words written on them and the artist wants the words to all be horizontal in the final mosaic. The word tiles come in two sizes: The smaller tiles are 4 inches tall and 4 inches wide, while the large tiles are 6 inches tall and 12 inches wide. If the small tiles cost $3.50 each and the larger tiles cost $4.50 each, how many of each should be used to minimize the cost? What is the minimum cost?

Answer 1

This is the answer: 0 small tile, 6 large tiles; minimum cost $27

Answer 2

i know the answer is : 0 small tile, 6 large tiles; minimum cost $27

but i got 0 small tiles, 30 large tiles; minimum cost of $135

Answer 3

here is my work could. Someone tell me were i went wrong?

The smallest size the mosaic can be is 3’ X 5’ or 36” X 60” (2160sq”)
The area of the small tiles are 4” X 4” or 16sq” at a cost of $3.50 each
The area of the large tiles are 6” X 12” or 72sq” at a cost of $4.50 each

Let x = the number of small tiles
Let y = the number of large tiles

(Area of the small tiles) (Number of small tiles) + (area of the large tiles) (number of large tiles) = minimum area of the mosaic
16x + 72y = 2160

Start by finding “y”
16x + 72y = 2160
16x + y = 30
y = 30 – 16x

Plug “y” back into the original equation to find x (the number of small tiles)
16x + 72y = 2160
16x + 72(30 – 16x) = 2160
16x + 2160 – 1152x = 2160
-1136x = 0
x = 0/-1136
x = 0
The number of small tiles (x) = 0

Plug “x” into the original equation to find “y” (the number of large tiles)
16x + 72y = 2160
16(0) + 72y = 2160
72y = 2160
y = 2160/72
y = 30
The number of large tiles (y) = 30

Minimum cost = (price of each small tile) (Number of small tiles) + (price of each large tile) (number of large tiles)
Minimum cost = $3.50x + $4.50y
Minimum cost = $3.50(0) + $4.50(30)
Minimum cost = $0 + $135
Minimum cost = $135

So in order to minimize the cost the artist should use 0 small tiles and 30 large tiles to keep a minimum cost of $135

Answer 4

@mojo872 can you help me please?

Answer 5

As far as I can see, all of your workings are correct - the artist lays out 30 large tiles (6 high and 5 wide), to create a mosaic that is 3 feet high and 5 feet wide, which means the answer is indeed $135.

Can you check the question wording again, and make sure all of the numbers you typed out are correct?

Answer 6

@mojo872 that’s exactly how the question is worded. I copied it from the test my teacher gave me. I sent in that answer and got 0 pts. Though.

I’ve emailed my teacher asking for an explanation … maybe there’s something missing from the question that she’s not giving me. I don’t know but I’ll wait and see. Thx for your help though :)

Answer 7

No problem. On reviewing your workings, I think you've made an incorrect algebraic manipulation here - when dividing by 72:

Start by finding “y”
16x + 72y = 2160
16x + y = 30
y = 30 – 16x

I still agree that the cheapest way is to use 30 large tiles (which seems obvious because they are 4.5 times bigger than the small tiles, and only cost $1 more), but I don't think your workings prove that, and I don't know how to prove it rigorously.

However, your answer is still correct, as far as I can see, so that doesn't explain getting zero! Let us know what your teacher says...

Answer 8

ok so my teacher finnally responded. here's what she said ... "You are missing a lot of other inequalities that play a huge factor in solving this problem. You need not only the area, but the height equation, the width equation, and the real world constraints. You only have two of the 6 that you need."

Answer 9

This problem is simpler than you think.

If you calculate the price per square inch of the small word tiles and the large word tiles, you'll see that the large word tiles are much cheaper per square inch. Therefore, it will be much cheaper to cover with all large word tiles.

The only problem that could occur is if the large tiles are of a size that is not divisible into the dimensions of the entire mosaic, but in this case they are, so just use large tiles.

4 ft x 6 ft = 24 sq ft = 2160 sq in.
6 in. x 12 in. = 72 sq in.

(2160 sq in. / 72 sq in. = 135 large tiles

Answer 10

but umm ... 2160 / 72 = 30 not 135 @mathstudent55

Answer 11

Sorry. I meant 30, not 135. I had just looked at 135 and wrote it by mistake. You are correct, it's 30 large tiles.

Answer 12

haha its ok ... I think my teachers crazy. I don’t see how the answer could possibly be 6 large tiles

Answer 13

This problem is simpler than you think.

If you calculate the price per square inch of the small word tiles and the large word tiles, you'll see that the large word tiles are much cheaper per square inch. Therefore, it will be much cheaper to cover with all large word tiles.

The only problem that could occur is if the large tiles are of a size that is not divisible into the dimensions of the entire mosaic, but in this case they are, so just use large tiles.

5 ft x 3 ft = 15 sq ft = 2160 sq in.
6 in. x 12 in. = 72 sq in.

(2160 sq in. / 72 sq in. = 30 large tiles

Answer 14

I also wrote above 4 ft x 6 fx = 24 sq ft = 2160 sq in.
In fact, 4 ft x 6 ft = 24 sq ft = 3456 sq in.

I meant 3 ft x 5 ft = 15 sq ft = 2160 sq in.
I corrected in my last response.

Answer 15

@mathstudent55 @mojo872 so what would the equation be for the real answer? how would you get that?