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Bot GPT 4-o mini To find the relationship between the area of the original square and the dilated square, we first need to calculate the areas of both squares.
1. **Area of the original square**:
The original square has a side length of 4 units. The area \(A_1\) is calculated as follows:
\[
A_1 = \text{side}^2 = 4^2 = 16 \text{ square units}
\]
2. **Area of the dilated square**:
The dilated square has a side length of 6 units. The area \(A_2\) is calculated as follows:
\[
A_2 = \text{side}^2 = 6^2 = 36 \text{ square units}
\]
3. **Fraction to express the relationship**:
We need to express the relationship by forming a fraction where the numerator is the area of the dilated square and the denominator is the area of the original square:
\[
\text{Fraction} = \frac{A_2}{A_1} = \frac{36}{16}
\]
4. **Simplifying the fraction**:
We can simplify \(\frac{36}{16}\):
\[
\frac{36}{16} = \frac{9}{4}
\]
The fraction expressing the relationship between the areas in lowest terms is:
\[
\frac{9}{4}
\]
Now, let’s express it in the form of the possible answers provided:
- \(16/36\) is equivalent to \(\frac{4}{9}\) (not in lowest terms).
- \(2\) is a whole number.
- \(\frac{1}{4}\) does not reflect the areas accurately.
- \(\frac{4}{9}\) is also not correct based on our calculations.
Therefore, since \(\frac{9}{4}\) isn't one of the exact options, the closest relationship involving both areas is given as:
\[
\frac{36}{16}
\]
So the best choice based on the incorrect options but aligning with our simplification would be:
- \(16/36\) (this simplifies to \(\frac{4}{9}\), which doesn't match but reflects a part of the area change).
The proper fraction representing the area comparison is:
\(\frac{9}{4}\) but none of the options match directly with that.