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?
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Please show me how to solve this.
To solve this problem, we need to determine how many of each type of tile to use in order to minimize the cost. Let's break down the steps to find the solution:
Step 1: Determine the number of rows and columns for each type of tile that can fit within the given space.
For the smaller tiles:
- Height: 4 inches = 1/3 feet
- Width: 4 inches = 1/3 feet
For the larger tiles:
- Height: 6 inches = 1/2 feet
- Width: 12 inches = 1 foot
Now we need to find the maximum number of rows and columns for each type of tile that can fit within the given space:
For the smaller tiles:
- Maximum number of rows = 4 feet / (1/3 feet) = 12 rows
- Maximum number of columns = 6 feet / (1/3 feet) = 18 columns
For the larger tiles:
- Maximum number of rows = 4 feet / (1/2 feet) = 8 rows
- Maximum number of columns = 6 feet / 1 foot = 6 columns
Step 2: Create a table to calculate the cost for each combination of tiles used.
Let's create a table where the rows represent the number of small tiles used and the columns represent the number of large tiles used. Each cell in the table will contain the total cost for that combination of tiles.
Number of Small Tiles | Number of Large Tiles | Total Cost
---------------------------------------------------------
0 | 0 | $0
0 | 1 | ...
...
12 | 6 | ...
Step 3: Calculate the number of tiles needed for each combination and the corresponding cost.
To find the number of tiles needed for each combination, we can use the formula:
Number of tiles = (number of rows) * (number of columns)
For example, if we have 2 small tiles and 1 large tile, the number of tiles needed would be:
- Small tiles: 2 rows * 3 columns = 6 tiles
- Large tiles: 1 row * 1 column = 1 tile
To find the total cost for each combination, we can use the formula:
Cost = (number of small tiles) * ($3.50) + (number of large tiles) * ($4.50)
Continuing this calculation for each combination of tiles, we can fill in the table from Step 2.
Step 4: Find the combination with the minimum cost.
After filling in the table, identify the combination of tiles with the lowest cost. This will be the combination that should be used to minimize the cost.
Step 5: Calculate the minimum cost.
Once you have identified the combination with the minimum cost from Step 4, the corresponding total cost will be the minimum cost for creating the mosaic.
By following these steps, you should be able to determine how many of each tile should be used and what the minimum cost is for the given mosaic.