To determine the area of the square tile for the seventh deck, we first need to calculate the area of the seventh deck based on its perimeter. The perimeter of a square is given by the formula:
\[ P = 4s \]
where \( s \) is the length of one side of the square. We can rearrange this formula to find \( s \):
\[ s = \frac{P}{4} \]
From the table, we can see that the perimeter of the decks follows an arithmetic sequence with a common difference of 4 feet.
Given the perimeters:
- 1st Deck: 24 feet
- 2nd Deck: 28 feet
- 3rd Deck: 32 feet
- 4th Deck: 36 feet
To find the perimeter of the 7th deck, we can continue the pattern:
- 5th Deck: \( 36 + 4 = 40 \) feet
- 6th Deck: \( 40 + 4 = 44 \) feet
- 7th Deck: \( 44 + 4 = 48 \) feet
Now, we can calculate the side length of the 7th deck:
\[ s = \frac{48 \text{ feet}}{4} = 12 \text{ feet} \]
Next, we can find the area of the square deck using the formula for the area of a square:
\[ A = s^2 = (12 \text{ feet})^2 = 144 \text{ square feet} \]
According to Connor's rule, the area of the tile is half the area of the deck:
\[ \text{Area of the tile} = \frac{A}{2} = \frac{144 \text{ square feet}}{2} = 72 \text{ square feet} \]
Thus, the area of the tile for the seventh deck is:
72 square feet.
2 answers