Let's go through the steps to determine the dimensions of the original photo.
1. Understand the Problem:
- Sylvia enlarged a photo to dimensions of [tex]\( 24 \, \text{inches} \times 32 \, \text{inches} \)[/tex].
- The dilation factor used is 4.
- We need to find the original dimensions of the photo before it was enlarged.
2. Use the Dilation Factor to Find the Original Dimensions:
- When a photo is enlarged by a dilation factor, each dimension of the photo is multiplied by this factor.
- Therefore, to find the original width, we will divide the enlarged width by the dilation factor.
- Similarly, to find the original height, we will divide the enlarged height by the dilation factor.
The equations are:
[tex]\[
\text{Original Width} = \frac{\text{Enlarged Width}}{\text{Dilation Factor}}
\][/tex]
[tex]\[
\text{Original Height} = \frac{\text{Enlarged Height}}{\text{Dilation Factor}}
\][/tex]
3. Substitute the Given Values:
- Enlarged Width is 24 inches.
- Enlarged Height is 32 inches.
- Dilation Factor is 4.
Calculating the original dimensions:
[tex]\[
\text{Original Width} = \frac{24}{4} = 6 \, \text{inches}
\][/tex]
[tex]\[
\text{Original Height} = \frac{32}{4} = 8 \, \text{inches}
\][/tex]
4. Verify with the Given Choices:
- The possible given choices are [tex]\(3 \times 8\)[/tex], [tex]\(6 \times 8\)[/tex], [tex]\(12 \times 16\)[/tex], and [tex]\(18 \times 24\)[/tex].
- From our calculations, the original dimensions of the photo are [tex]\(6 \, \text{inches} \times 8 \, \text{inches}\)[/tex], which matches one of the provided choices.
Therefore, the dimensions of the original photo are [tex]\( 6 \times 8 \)[/tex] inches.
