To determine the values of \( x \) and \( y \) such that the two houses are similar, we need to ensure that the corresponding sides of the houses maintain the same ratio.
From the information given, we'll denote the dimensions for both houses as follows:
House One:
- Top: 6
- Side: 12
- Bottom: 10
- Inside: 120, 120 (same for both houses)
House Two:
- Top: \( y \)
- Side: \( x \)
- Bottom: 30
- Inside: 120, 120 (same for both houses)
Since the inside values are the same (120 in both houses), we can focus on matching the other dimensions in the ratio.
Set Up Ratios: To find the values of \( x \) and \( y \) that make the two houses similar, we'll set up proportions based on the corresponding sides:
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Top sides: \[ \frac{y}{6} = k \]
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Side sides: \[ \frac{x}{12} = k \]
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Bottom sides: \[ \frac{30}{10} = k \]
From the bottom sides, we have: \[ \frac{30}{10} = 3 \implies k = 3 \]
Now substituting \( k \) back into the ratios for \( y \) and \( x \):
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For the top: \[ \frac{y}{6} = 3 \implies y = 6 \cdot 3 = 18 \]
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For the sides: \[ \frac{x}{12} = 3 \implies x = 12 \cdot 3 = 36 \]
Therefore, the values of \( x \) and \( y \) that make the two houses similar are: \[ x = 36 \quad \text{and} \quad y = 18 \]
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