Question

Connor is building square decks. In the center of each deck, Connor places a square tile. The perimeter of the first 4 decks are shown in the table below. If Connor's rule to find the area of the square tile is to divide the area of the deck by 2, what would be the area of the tile for the seventh deck? Deck Perimeter First 24 feet Second 28 feet Third 32 feet Fourth 36 feet (1 point) Responses 6 square feet 6 square feet 12 square feet 12 square feet 72 square feet 72 square feet 144 square feet 144 square feet Skip to navigation

Answer 1

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:

1. **First Deck (Perimeter = 24 feet)**:
\[
s = \frac{24}{4} = 6 \quad \Rightarrow \quad A = 6^2 = 36 \text{ square feet}
\]

2. **Second Deck (Perimeter = 28 feet)**:
\[
s = \frac{28}{4} = 7 \quad \Rightarrow \quad A = 7^2 = 49 \text{ square feet}
\]

3. **Third Deck (Perimeter = 32 feet)**:
\[
s = \frac{32}{4} = 8 \quad \Rightarrow \quad A = 8^2 = 64 \text{ square feet}
\]

4. **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**.