To find the area of the square tile for the seventh deck, we first need to calculate the area of each deck given its perimeter.
The formula for the perimeter \( P \) of a square is:
\[ P = 4s \]
where \( s \) is the length of one side of the square. We can rearrange this formula to find the side length:
\[ s = \frac{P}{4} \]
Next, we can find the area \( A \) of the square deck using the formula:
\[ A = s^2 = \left(\frac{P}{4}\right)^2 \]
Now, let's calculate the area for the first four decks based on their perimeters:
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First Deck (Perimeter = 24 feet): \[ s = \frac{24}{4} = 6 \quad \Rightarrow \quad A = 6^2 = 36 \text{ square feet} \]
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Second Deck (Perimeter = 28 feet): \[ s = \frac{28}{4} = 7 \quad \Rightarrow \quad A = 7^2 = 49 \text{ square feet} \]
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Third Deck (Perimeter = 32 feet): \[ s = \frac{32}{4} = 8 \quad \Rightarrow \quad A = 8^2 = 64 \text{ square feet} \]
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Fourth Deck (Perimeter = 36 feet): \[ s = \frac{36}{4} = 9 \quad \Rightarrow \quad A = 9^2 = 81 \text{ square feet} \]
Now, following Connor's rule, the area of the tile is half the area of the deck:
- First Deck Tile Area: \(\frac{36}{2} = 18\) square feet
- Second Deck Tile Area: \(\frac{49}{2} = 24.5\) square feet
- Third Deck Tile Area: \(\frac{64}{2} = 32\) square feet
- Fourth Deck Tile Area: \(\frac{81}{2} = 40.5\) square feet
Now, we need to find the perimeter of the seventh deck. From the previous perimeters (24, 28, 32, 36), it can be observed that the perimeter increases by 4 feet each time. Thus:
- Fifth Deck: 36 + 4 = 40 feet
- Sixth Deck: 40 + 4 = 44 feet
- Seventh Deck: 44 + 4 = 48 feet
Now we can calculate the area for the seventh deck:
\[ s = \frac{48}{4} = 12 \quad \Rightarrow \quad A = 12^2 = 144 \text{ square feet} \]
Using Connor's rule to find the area of the tile:
\[ \text{Area of Tile for Seventh Deck} = \frac{144}{2} = 72 \text{ square feet} \]
Thus, the area of the tile for the seventh deck is 72 square feet.
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