To solve the problem, let's break it down into clear, step-by-step instructions for determining the dimensions of the original photo.
1. Given Information:
- The enlarged photo has dimensions of [tex]\( 24 \)[/tex] inches in width and [tex]\( 32 \)[/tex] inches in height.
- The dilation factor used to enlarge the photo is [tex]\( 4 \)[/tex].
2. Understanding Dilation:
- Dilation involves resizing an object by a certain factor. A dilation factor of [tex]\( 4 \)[/tex] means each dimension of the original photo was multiplied by [tex]\( 4 \)[/tex] to obtain the dimensions of the enlarged photo.
3. Calculating the Original Dimensions:
- To find the original dimensions, we need to reverse the dilation process by dividing the dimensions of the enlarged photo by the dilation factor.
- Original Width:
[tex]\[
\text{Original Width} = \frac{\text{Enlarged Width}}{\text{Dilation Factor}} = \frac{24}{4} = 6 \text{ inches}
\][/tex]
- Original Height:
[tex]\[
\text{Original Height} = \frac{\text{Enlarged Height}}{\text{Dilation Factor}} = \frac{32}{4} = 8 \text{ inches}
\][/tex]
4. Result:
- Therefore, the dimensions of the original photo are [tex]\( 6 \)[/tex] inches by [tex]\( 8 \)[/tex] inches.
5. Verification:
- If we multiply these original dimensions by the dilation factor ([tex]\( 4 \)[/tex]), we should get back the dimensions of the enlarged photo:
[tex]\[
\text{New Width} = 6 \times 4 = 24 \text{ inches}
\][/tex]
[tex]\[
\text{New Height} = 8 \times 4 = 32 \text{ inches}
\][/tex]
- This confirms that our calculations are correct.
Thus, the correct answer is [tex]\( \boxed{6 \times 8} \)[/tex].
