The side of a grassy square plot is 80 m. Two cross paths each 4m wide are constructed at right angles through the centre of the plot such that each path is parallel to the side of the plot. Determine the area of the paths.
Answers (1)
The side S of a grassy square plot is 80m. Thus, the area A amounts to:
A = S^2 = 80^2m^2 = 6400m^2
The paths are 4m wide and cut through this large grassy plot
dividing them into 4 identical small grassy square plots with a side length s of
s = (S - 4) / 2 = (80m - 4m) / 2 = 38m
The area a of one small square amounts to:
a = s^2 = 38^2m^2 = 1444m^2
There are 4 of them:
4a = 4 * 1444m^2 = 5776m^2
The difference A - 4a equals the area p of the cross paths:
p = A - 4a = 6400m^2 - 5776m^2 = 624m^2.
Well, let's approach this in a more elegant, mathematical way casting aside for the moment any consideration as to horticultural aesthetics. After all, we're just interested in areas.
S = 80m ; side of the squared grassy plot
W = 4m ; width of the paths
But now we move the paths to the border of the plot, the vertical path to, say, the right border and the horizontal path to the bottom. We get a large square with side length
a = S - W = 76m and a small square in the bottom right corner with side length
b = W = 4m
The whole grassy plot has an area of
(a + b)^2m^2 = (a^2 + 2ab + b^2)m^2. And I'm quite confident that this should look familiar to you.