To understand why Allen is incorrect, let's first clarify the concept of scaling areas compared to scaling linear dimensions (lengths, widths, etc.).
When you scale a shape, you are increasing or decreasing its size by a certain factor. If you scale the lengths of a shape, you multiply the linear dimensions by the scale factor. If a scale factor is less than 1, you are reducing the size, and if it is greater than 1, you are increasing it. However, when you scale areas, the scale factor affects the area differently than it does the lengths.
For areas, the factor that applies is the square of the linear scale factor. This is because the area of a two-dimensional shape is proportional to the square of its dimensions.
Here are the possible reasons why Allen might be incorrect:
1. Allen is incorrect because he applied the scale to the sides and then found the total area of the woods. This implies that Allen multiplied each side of the woods by the scale factor and then computed the area. If he did this, he would have used the linear scale factor directly on the area, which is incorrect. The area should be multiplied by the square of the scale factor instead.
2. Allen is incorrect because he found the total area of the woods and then applied the scale. This could be correct or incorrect depending on how he applied the scale factor. If he applied the linear scale factor to the area, then he is wrong. If he squared the scale factor and then applied it to the area, he would be correct.
3. Allen is incorrect because he did not apply the scale. This statement is wrong because it doesn't provide enough information about what Allen did or didn't do. Merely not applying the scale doesn't specify the mistake.
4. Allen is incorrect because he used the formula to find perimeter instead of area. This statement is incorrect regarding the question of scale factors, as the mistake described doesn't involve scaling at all but rather using an entirely wrong formula.
Based on the context given, the most likely reason Allen is incorrect is the first one:
Allen is incorrect because he applied the scale to the sides and then found the total area of the woods. This indicates that Allen scaled the linear dimensions and then calculated the area with the scaled dimensions, which would not give the correct scaled area. Instead, he should have calculated the area first with the original dimensions and then applied the square of the scale factor to scale the area correctly.
