a rectangle of integer dimensions is filed with 1x1 tiles. Each tile that touches the perimeter is colored red, while all the "interior" tiles are left white. Let R be the number of red tiles and W the number of whites tiles
find R and W for a 3x10 rectangle
find any and all rectangles for which R and W are equal
Find any and all rectanglef for which R is twice the size of W
Find any and all rectanlges for which E is twice the size of R
for the 3x10
total squares = 30
inner W's = 1x8 = 8
Reds = 30-8 = 12
In general
let width be x, and length be y
total squares = xy
W = (x-2)(y-2) = xy -2x -2y + 4
then R = xy - (xy - 2x - 2y +4)
= 2x + 2y - 4
for R = W
2x+2y-4 = xy - 2x - 2y + 4
4x+4y-8=xy
4y - xy = 8-4x
y(4-x) = 8-4x
y = (8-4x)/(4-x) = 4(x-2)/(x-4)
so pick any value of x , x>4 , and y an integer.
e.g. x = 8, then y = 6
x=12, y= 5
x = 60 , y ≠ integer
since the function approaches 4 as x gets larger, and we are already at y = 5 when x=12
the only two cases are :
8 by 6 or 12 by 5
for R = 2W , try it yourself
for the last one, I don't know what E is . Did you mean W ?
To find the number of red (R) and white (W) tiles in a rectangle, we need to understand the pattern based on the dimensions.
For the given 3x10 rectangle, we can calculate the number of red and white tiles using the following steps:
Step 1: Calculate the number of red tiles:
- Each row on the top and bottom will have red tiles, which is equal to the width of the rectangle. In this case, the width is 10, so each row will have 10 red tiles: (10 x 2) = 20 red tiles on the top and bottom.
- The remaining rows (interior rows) do not have any red tiles.
Step 2: Calculate the number of white tiles:
- To find the total number of tiles, we multiply the length by the width. In this case, 3 x 10 = 30 tiles.
- Subtract the number of red tiles from the total number of tiles to get the number of white tiles: 30 - 20 = 10 white tiles.
Therefore, for the 3x10 rectangle, R = 20 and W = 10.
Now let's move on to the other cases:
Case 1: Rectangles where R = W (equal number of red and white tiles)
When R = W, it means that the rectangle has an even number of rows and columns because the red tiles are placed symmetrically on each side.
For example:
- A 2x2 rectangle will have 4 red tiles and 0 white tiles.
- A 3x3 rectangle will have 12 red tiles and 1 white tile.
- A 4x4 rectangle will have 16 red tiles and 4 white tiles.
- And so on.
In general, for a rectangle of size nxn, R = n^2 - (n-2)^2, and W = (n-2)^2.
Case 2: Rectangles where R is twice the size of W
When R is twice the size of W, it means that the number of red tiles is double the number of white tiles.
For example:
- A 2x3 rectangle will have 10 red tiles and 1 white tile.
- A 3x5 rectangle will have 28 red tiles and 4 white tiles.
- A 4x7 rectangle will have 52 red tiles and 9 white tiles.
- And so on.
In general, for a rectangle of size nxm, where n > m:
- R = (2n + 2m) - 4
- W = (n-2)(m-2)
Case 3: Rectangles where E (exterior tiles) is twice the size of R
Unfortunately, from the given question, there is no information or pattern provided concerning E (exterior tiles) being twice the size of R. Therefore, we cannot provide any specific formulas or examples for that scenario.
By applying these patterns and calculations, you can determine the number of red and white tiles for different rectangle dimensions.