Question

linear programming method

Answer 1

What is the question?

Answer 2

we have to follow linear inequality to for a equation

Answer 3

I am here! I hate word problems but I am trying

Answer 4

Let there be x small tiles and y large tiles.
Can you find the area of small tiles and large tiles? and then sum it to find the total area of the mosaic ? Let it be M

Answer 5

@LilySwan

Answer 6

area of a mosaic is 88= 16+72

Answer 7

Oh one thing
there are x small tiles so 16x is the area contributed by small tiles
and there are y large tiles so 72y is the area contributed by large tiles
so total area of mosaic
\[M=16x+72y\]
Now there is a condition on the area of the mosaic, what's that? @LilySwan

Answer 8

The mosaic must be at least 3 feet tall and 5 feet wide.

Answer 9

and one more?

Answer 10

The tiles in the mosaic have words written on them and the artist wants the words to all be horizontal in the final mosaic

Answer 11

"An artist is creating a mosaic that cannot be larger than the space allotted which is 4 feet tall and 6 feet wide."
So we have a limit on the area of mosaic
\[3 \times 5\le M \le 4\times 6\]
\[15 \le M \le 24\]
Do you get this part?

Answer 12

yep

Answer 13

so
\[15 \le 16x+72y \le 24\]

Now what would be the cost C of x small and y large tiles?

Answer 14

I would stick to the same units: inches perhaps ?

Answer 15

Thanks @phi We missed that, so we can multiply both the limits by 12 x 12 to get units in inch^2
\[144 \times 15\le 16x+72y \le 24 \times 144\]
\[2160 \le 16x+72y \le 3456\]
Do you get this part @LilySwan

Answer 16

yep

Answer 17

C= 3.5 x +4.5 y

Answer 18

So we have two conditions:
\[2160 \le 16x+72y \le 3456\]
\[C=3.5x+4.5y\]
let's think how we can find the minimum cost

Answer 19

ok

Answer 20

Let's multiply cost equation by 16
\[16C=56x+72y\]
We get
\[72y=16C-56x\]
so
\[2160\le 16C-56x \le 3456\]
Do you get this?

Answer 21

yep

Answer 22

I'm thinking how do we proceed now?

Answer 23

you have any ideas?

Answer 24

btw
\[ 2160 \le 16x+72y \le 3456 \\ 2160 \le 16x + 16C-56x \le 3456 \\ 2160 \le 16C-40x \le 3456 \]

Answer 25

oh where is my mind,
\[2160 \le 16C-40x \le 3456\]

@phi please help what should we do now?

Answer 26

make C a function of x : we get 2 boundary lines

\[ C ≤ \frac{40}{16}x + \frac{3456}{16} \]
and a lower bound