Question

A square fire pit with a side length of 's' feet is bordered by 1-foot square stones as shown.

(a) How many stones does it take to border the fire pit with two rows of stones? Use a diagram to justify your answer.

(b) You border the fire pit with rows of stones. How many stones are in the nth row? Explain your reasoning.

Answer 1

Hint: Area of square = $sid{e}^2$

Step 1: Part (a)
Given each border is of width 1ft
So the sides of the complete square including both borders and inner square is= $(s+4)$ ft.
And the area of both borders
= [Area of complete square having each side $(s+4)ft$ft. – Area of inner square having side s ft.]
= $[{(s+4)}^2-s^2]$] square feet.

Step 2: Now no. stones of 1-foot square required is:
Area of each stone is 1square feet.
$$\begin{gathered} Numberofstones=\frac{totalareaofborders}{areaofeachstone} \\ =\frac{[{(s+4)}^2-s^2]squarefeet}{1squarefeet} \\ Numberofstones=[{(s+4)}^2-s^2] \end{gathered}$$

Step 3: Part (b).
If there are n rows then the length of complete square is = $(s+2n)$

And the side of the square formed till (n – 1) rows = $[s+2(n-1\left)\right]$

Area of the border of nth row = (Area of the complete square till nth row) – ( Area of the square till $(n-1)$row
Area of nth border = ${(s+2n)}^2-{[s+2(n-1\left)\right]}^2$

So again no of stones in nth row border are same as the area

${(s+2n)}^2-{[s+2(n-1\left)\right]}^2$