Question: Problem 2
Here are the first three steps in a pattern.
How many tiles will there be when $s = 10$?
tiles
Edit my response
Explain your thinking.
$s = 1$
$s = 2$
$s = 3$
Graph shapes: three purple tile arrays on the left side of the page, labeled $s = 1$, $s = 2$, and $s = 3$; the first array is a $1\times 2$ rectangle, the second is a $2\times 3$ rectangle, and the third is a $3\times 4$ rectangle; position_hint = bottom-left
Select all of the expressions that could represent the number of
tiles in this diagram.
$5 \cdot 7$
$5 \cdot 7 - 6$
$3 \cdot 7 + 2 \cdot 2$
$5 \cdot 7 - 2 \cdot 3$
$2 \cdot 5 + 3 \cdot 3 + 2 \cdot 5$
Graph shape: a pink tile diagram made of a $5$-by-$7$ outer rectangle with a centered rectangular cutout from the bottom, leaving two $2$-by-$2$ legs; position_hint \in center
1. **State the problem:** We have a pattern of tile arrays where for each step $s$, the array is a rectangle of size $s \times (s+1)$. We want to find the number of tiles when $s = 10$.
2. **Identify the pattern:** From the given steps:
- When $s=1$, tiles = $1 \times 2 = 2$
- When $s=2$, tiles = $2 \times 3 = 6$
- When $s=3$, tiles = $3 \times 4 = 12$
3. **General formula:** The number of tiles for step $s$ is:
$$\text{tiles} = s \times (s+1)$$
4. **Calculate for $s=10$:**
$$\text{tiles} = 10 \times (10 + 1) = 10 \times 11 = 110$$
5. **Answer:** There will be **110 tiles** when $s=10$.
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6. **Analyze the expressions for the pink tile diagram:**
The pink tile diagram is a $5 \times 7$ rectangle with a rectangular cutout in the center bottom, leaving two $2 \times 2$ legs.
7. **Calculate total tiles in the $5 \times 7$ rectangle:**
$$5 \times 7 = 35$$
8. **Calculate the cutout area:**
The cutout is a rectangle of size $3 \times 4$ (since the legs are $2 \times 2$ each, the cutout width is $7 - 2 - 2 = 3$ and height $5 - 2 = 3$), but the problem states the cutout is centered at the bottom, so the cutout area is $3 \times 4 = 12$ tiles.
9. **Calculate remaining tiles:**
$$35 - 12 = 23$$
10. **Check the expressions:**
- $5 \cdot 7 = 35$ (total tiles without cutout)
- $5 \cdot 7 - 6 = 29$ (incorrect cutout)
- $3 \cdot 7 + 2 \cdot 2 = 21 + 4 = 25$ (does not match 23)
- $5 \cdot 7 - 2 \cdot 3 = 35 - 6 = 29$ (incorrect cutout)
- $2 \cdot 5 + 3 \cdot 3 + 2 \cdot 5 = 10 + 9 + 10 = 29$ (incorrect total)
None of these expressions exactly equal 23, so none perfectly represent the number of tiles in the pink diagram as described.
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**Final answer:**
- Number of tiles when $s=10$ is **110**.
- None of the given expressions exactly represent the pink tile diagram's tile count based on the description.