Question

ATTENTION I ALREADY KNOW THE ANSWER I JUST WANT SOMEONE TO HELP ME WORK THE PROBLEM STEP BY STEP An artist is creating a mosaic that cannot be larger than the space allotted which is 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

@phi can you please help

Answer 2

we need some "relations" that tell us how many big tiles we can use
and how many small tiles we can use.

Answer 3

for example, the big tiles (measured in feet) or 0.5 tall and 1 wide

if the mosaic must be 3 feet tall, how many big tiles do we need to make a height of 3 feet?

Answer 4

6?

Answer 5

yes.

Answer 6

Ok, what is next?

Answer 7

well this is what I got as the answer 0 small tile, 6 large tiles; minimum cost $27 so now i just need to know how to work the problem all together

Answer 8

This is a problem in linear programming.

The area of a big block is ½ sq ft
the area of a small block is 1/9 sq ft

if we have x small blocks and y big blocks, the area is
x/9 + y/2

this area must be larger than the 3 x 5 ft min= 15 sq ft
but smaller than the large region 4 x 6 = 24

\[ 15≤ \frac{x}{9}+ \frac{y}{2} ≤ 24 \]
we would plot those two lines as a start

Answer 9

ok, I am with you

Answer 10

we will get an "allowed region" between the two lines.

we next need to plot lines showing the allowed max number of blocks.

Answer 11

ok, can you show me?please

Answer 12

Here are the two lines

Answer 13

I think the max is 6

Answer 14

nevermind i think i got it thank you

Answer 15

ok

Answer 16

Here is the plot with the intersections at points A, B , C and D

we optimum point will be one of these. We evaluate the cost of each of these points

Answer 17

the cost of point A (0,48) (where 48 is the number of big blocks) is
4.50*48= $216

the cost of point B is
4.50*30= $135

the cost of point C (135,0) is
135*3.50= $472.50

and the cost of point D (216,0) is
216*3.50= $756