### Solution:

Let us re-draw the given triangular flower bed border.

Assume the length of one piece of rectangular plate is "x" inches.

We know that the sides are cut from these "x" inches. To get 3 rectangular boards for the triangular edge of the bed, we need to cut this single piece twice.

So total length of wasted board = inches

We know that 1 foot = 12 inches.

So length of one piece of rectangular plate (x) = AB + BC + CA + lost length.

$\n \n \n \n \u21d2x=(4\xd712)+(6\xd712)+(5\xd712)+\n 1\n 4\n \n \n \n \n \n \n \u21d2x=48+72+60+0.25\n \n \n \n \n $

$\n \n \u21d2x=180.25\n $ inches.

We find the nearest whole number that is greater than 180.25 inches, which is 181 inches.

So we found the shortest single piece of board that Manu can use as 181 inches.

So the correct answer is option 4.