A house has a backyard with a square fire pit with side length

Question

feet, surrounded by two rows of 1-foot square stones.

The house also has a rectangular patio area where the length is 8 more feet than the width,
. The patio is surrounded with 4-foot square stones.

Multiple Choice Question
Which expression represents the number of stones needed for the two rows surrounding the square fire pit?

Answer 1

To find the expression that represents the number of stones needed for the two rows surrounding the square fire pit with a side length of \( s \) feet, we can start with these calculations:

1. Determine the total area of the fire pit itself: The fire pit is a square with side length \( s \), so its area is: \[ s^2 \]

2. Dimensions of the outer square including two rows of stones: Since there are two rows of stones surrounding the fire pit, which are 1-foot wide each, we need to add 2 feet to each side of the fire pit to get the outer dimensions. Therefore, the outer side length of the square, including the stones, is: \[ s + 2 \times 1 = s + 2 \text{ feet} \]

3. Calculate the total area of the outer square: The area of the larger square (fire pit plus the surrounding stones) is: \[ (s + 2)^2 \]

4. Calculate the area of the stones: The area covered by the stones is simply the area of the outer square minus the area of the fire pit: \[ \text{Area of stones} = (s + 2)^2 - s^2 \]

5. Expand this expression using the difference of squares: \[ (s + 2)^2 - s^2 = (s^2 + 4s + 4) - s^2 = 4s + 4 \]

Conclusion: The expression that represents the number of stones needed for the two rows surrounding the square fire pit is: \[ 4s + 4 \]