a grid has line at 90-degree angles. There are 12 lines in one direction and 9 lines in the other direction. Lines that are parallel are 11 inches apart. What is the least number of 12-inch floor tiles needed to cover all of the line interestions on the grid?
To find the least number of 12-inch floor tiles needed to cover all of the line intersections on the grid, we need to calculate the total area of the grid and then divide it by the area covered by each tile.
First, let's calculate the number of line intersections on the grid. We have 12 lines in one direction and 9 lines in the other direction. Since each line intersects with every line in the other direction, we can multiply the number of lines in each direction to get the total number of intersections: 12 * 9 = 108 intersections.
Next, we need to calculate the area covered by each tile. The tiles are 12 inches long and 12 inches wide, so they cover an area of 12 * 12 = 144 square inches.
Now, we can calculate the total area of the grid. If the lines that are parallel are 11 inches apart, it means we have 11 inches between each intersection in both directions. Therefore, the grid is (12 - 1) * 11 inches wide and (9 - 1) * 11 inches tall.
The width of the grid is (12 - 1) * 11 = 11 * 11 = 121 inches.
The height of the grid is (9 - 1) * 11 = 8 * 11 = 88 inches.
To find the total area of the grid, we multiply the width by the height: 121 * 88 = 10,648 square inches.
Finally, we divide the total area of the grid by the area covered by each tile to find the least number of tiles needed.
Number of tiles = Total area / Area covered by each tile
Number of tiles = 10,648 / 144
Number of tiles ≈ 73.89
Since we cannot have a fraction of a tile, we need to round up to the nearest whole number. Therefore, the least number of 12-inch floor tiles needed to cover all of the line intersections on the grid is 74 tiles.