To compare the areas of Soshi's and Jack's rhombuses, we need to first calculate the area of Soshi's rhombus.
The formula for the area \( A \) of a rhombus is:
\[ A = \text{base} \times \text{height} \]
Calculating Soshi's Rhombus Area
- Base = 12 in.
- Height = 10 in.
Using the formula:
\[ A_{\text{Soshi}} = 12 , \text{in} \times 10 , \text{in} = 120 , \text{in}^2 \]
Calculating Jack's Rhombus Area
Jack's rhombus has base and height measures that are double those of Soshi's rhombus.
- Base of Jack's rhombus = \( 2 \times 12 , \text{in} = 24 , \text{in} \)
- Height of Jack's rhombus = \( 2 \times 10 , \text{in} = 20 , \text{in} \)
Now, calculate the area of Jack's rhombus:
\[ A_{\text{Jack}} = 24 , \text{in} \times 20 , \text{in} = 480 , \text{in}^2 \]
Comparing the Areas
To compare the areas of Jack's and Soshi's rhombuses:
\[ \frac{A_{\text{Jack}}}{A_{\text{Soshi}}} = \frac{480 , \text{in}^2}{120 , \text{in}^2} = 4 \]
This means that Jack's rhombus has an area that is 4 times greater than that of Soshi's rhombus.
Explanation
Jack's rhombus has dimensions that are double those of Soshi's. Since the area is proportional to the product of the base and height, when both the base and height are multiplied by 2, the area increases by a factor of \( 2 \times 2 = 4 \). Therefore, Jack's rhombus has an area that is 4 times greater than Soshi's rhombus.
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