a landscaper uses square tiles to make garden paths the ends of the paths are always 2 tiles wide, and the center portions of the paths are 3 tiles wide. what equation gives the total number y of tiles needed to make a path with a length of x tiles. the landscaper has 100 tiles. what is the length of the longest path that the landscaper can make
x = total length
x-2 = center part
2 = end parts
y = 2*2 + 3(x-2)
y = 4 + 3x - 6
y = 3x - 2
If you have 100 tiles,
100 = 3x-2
102 = 3x
x = 34
So, a path of length 34 uses 4+96 = 100 tiles
To determine the equation for the total number of tiles needed to make a path with a length of x tiles, let's break down the path into its components:
- The ends of the path, which are always 2 tiles wide.
- The center portions of the path, which are 3 tiles wide.
Let's analyze how many 3-tile center portions we can fit in a path of length x. For every center portion, we need 3 tiles. If we consider the number of 3-tile center portions as (x - 4), we can calculate the number of tiles needed in the center portions by multiplying it by 3.
Additionally, we always need 2 tiles for the ends of the path.
Therefore, the equation for the total number of tiles needed can be expressed as:
y = 2 + 3(x - 4)
Simplifying the equation:
y = 2 + 3x - 12
y = 3x - 10
To find the length of the longest path that the landscaper can make, we need to determine the maximum value of x, given that the landscaper has 100 tiles.
Since we have the equation y = 3x - 10, and we know that y (the total number of tiles) is 100, we can substitute y = 100 into the equation:
100 = 3x - 10
Solving for x:
3x = 110
x = 110/3
However, since the length of the path needs to be a whole number, we can round x down to the nearest whole number:
x = 36
Therefore, the length of the longest path that the landscaper can make is 36 tiles.